Elementary Mathematics General Ability Test
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Integral Calculus and Differential Equations

Practice MCQs

Integral Calculus and Differential Equations form an important part of higher elementary mathematics. Integral Calculus studies integration as the inverse process of differentiation and is used to evaluate areas, accumulated quantities, and definite values. Differential Equations study equations involving derivatives and are widely used in growth, decay, motion, population, finance, science, and engineering applications.

Elementary Mathematics Integral Calculus Differential Equations Competitive Exams

Integral Calculus and Differential Equations form an important part of higher elementary mathematics. Integral Calculus studies integration as the inverse process of differentiation and is used to evaluate areas, accumulated quantities and definite values. Differential Equations study equations involving derivatives and are widely used in growth, decay, motion, population, finance, science and engineering applications.

Overview

Integral Calculus is the branch of calculus that deals with finding a function when its derivative is known. It is also used to calculate areas, accumulated values and total change over an interval.

Differential Equations are equations that contain derivatives. They describe how one quantity changes with respect to another quantity. Many real-world processes such as population growth, radioactive decay, cooling, interest growth models, motion and biological changes can be represented using differential equations.

Quick idea: Differentiation finds rate of change. Integration reverses this process and finds accumulated value. Differential equations connect a quantity with its rate of change.
Area What It Covers Exam Focus
Indefinite Integration Integration as inverse of differentiation. Standard integrals and methods.
Methods of Integration Substitution and integration by parts. Choosing the correct method.
Standard Integrals Algebraic, trigonometric, exponential and hyperbolic functions. Formula recall and direct application.
Definite Integrals Evaluation between limits. Area and accumulated value.
Areas Bounded by Curves Area under curves and area between curves. Application of definite integrals.
Differential Equations Order, degree, formation, general and particular solution. First order and first degree equations.
Growth and Decay Rate proportional to quantity. Population growth and exponential decay.

“Integration measures accumulation, while differential equations describe changing systems.”

Calculus Tip
Key points
  • Integration is the inverse of differentiation.
  • Indefinite integrals include \(C\).
  • Definite integrals do not include \(C\).
  • Substitution simplifies composite expressions.
  • Integration by parts is useful for products of functions.
  • Definite integrals help find areas.
  • Differential equations contain derivatives.
  • Order and degree must be identified carefully.
  • Growth and decay are common applications.
integration definite integrals area differential equations growth decay
Core Concept Bank
Indefinite Integral
\[ \int f(x)\,dx = F(x)+C \] where \(F'(x)=f(x)\).
Definite Integral
\[ \int_a^b f(x)\,dx = F(b)-F(a) \]
Differential Equation
An equation involving derivatives: \[ \frac{dy}{dx}+y=x \]
Growth / Decay Model
\[ \frac{dy}{dt}=ky \] Solution: \[ y=Ce^{kt} \]

Tip: If a question asks for accumulated value, area, or total effect, integration is usually involved.

Integration and Differential Equation Method Selection Guide

Integral Calculus and Differential Equation questions become easier when the student first identifies whether the problem is asking for antiderivative, area, accumulated value, order, degree, solution, growth or decay. The table below helps choose the correct method quickly.

Question Type What to Use Typical Clue
Direct integration Standard integral formulas \(\int x^n dx\), \(\int \sin x dx\), \(\int e^x dx\)
Indefinite integral Antiderivative with constant \(C\) No upper and lower limits are given
Definite integral Upper value minus lower value Limits such as \(\int_a^b f(x)\,dx\)
Composite expression Substitution method Function and its derivative appear together
Product of functions Integration by parts \(x e^x\), \(x\sin x\), \(x\log x\)
Area under a curve Definite integral Area bounded by curve and x-axis
Area between two curves Upper curve minus lower curve Two curves enclose a region
Order of differential equation Highest order derivative present \(\frac{dy}{dx}\), \(\frac{d^2y}{dx^2}\), \(\frac{d^3y}{dx^3}\)
Degree of differential equation Power of highest order derivative Equation is polynomial in derivatives
Formation of differential equation Differentiate and eliminate constants Relation contains arbitrary constants
Variable separable equation Separate \(x\)-terms and \(y\)-terms, then integrate \(\frac{dy}{dx}=f(x)g(y)\)
Linear differential equation Integrating factor method \(\frac{dy}{dx}+Py=Q\)
Growth and decay Exponential model \(\frac{dy}{dt}=ky\) or \(\frac{dy}{dt}=-ky\)
Exam shortcut: First decide whether the question belongs to integration or differential equations. For integration, identify standard, substitution, parts or definite integral. For differential equations, identify order, degree and type before solving.

Integration as Inverse of Differentiation

Integration is the reverse process of differentiation. If the derivative of \(F(x)\) is \(f(x)\), then the integral of \(f(x)\) is \(F(x)+C\). Here, \(C\) is called the constant of integration.

Differentiation Integration Meaning
\(\frac{d}{dx}(x^3)=3x^2\) \(\int 3x^2\,dx=x^3+C\) Integration reverses differentiation.
\(\frac{d}{dx}(\sin x)=\cos x\) \(\int \cos x\,dx=\sin x+C\) Standard trigonometric integral.
\(\frac{d}{dx}(e^x)=e^x\) \(\int e^x\,dx=e^x+C\) Exponential function remains same.
\(\frac{d}{dx}(\log x)=\frac{1}{x}\) \(\int \frac{1}{x}\,dx=\log |x|+C\) Reciprocal function gives logarithm.
Important: In indefinite integration, always add \(C\), the constant of integration.

Standard Integrals

Standard integrals should be memorized because they are repeatedly used in direct integration, substitution, integration by parts, definite integrals, and differential equations.

Type Integral Result
Constant \(\int a\,dx\) \(ax+C\)
Power \(\int x^n\,dx\) \(\frac{x^{n+1}}{n+1}+C,\ n \neq -1\)
Reciprocal \(\int \frac{1}{x}\,dx\) \(\log |x|+C\)
Exponential \(\int e^x\,dx\) \(e^x+C\)
General Exponential \(\int a^x\,dx\) \(\frac{a^x}{\log a}+C\)
Sine \(\int \sin x\,dx\) \(-\cos x+C\)
Cosine \(\int \cos x\,dx\) \(\sin x+C\)
Tangent \(\int \tan x\,dx\) \(\log |\sec x|+C\)
Cotangent \(\int \cot x\,dx\) \(\log |\sin x|+C\)
Secant Square \(\int \sec^2 x\,dx\) \(\tan x+C\)
Cosecant Square / csc Square \(\int \csc^2 x\,dx\) \(-\cot x+C\)
Secant-Tangent \(\int \sec x \tan x\,dx\) \(\sec x+C\)
Cosecant-Cotangent / csc-cot \(\int \csc x \cot x\,dx\) \(-\csc x+C\)
Hyperbolic Sine \(\int \sinh x\,dx\) \(\cosh x+C\)
Hyperbolic Cosine \(\int \cosh x\,dx\) \(\sinh x+C\)
MathJax note: This code uses \(\csc x\) instead of the longer cosecant notation, because \(\csc\) is better supported by MathJax.

Integration by Substitution

Integration by substitution is useful when the integrand contains a composite expression and its derivative. The main idea is to replace a complicated expression by a simpler variable.

Step Action Example
Step 1 Choose a suitable substitution. Let \(u=x^2+1\).
Step 2 Differentiate the substitution. \(du=2x\,dx\).
Step 3 Rewrite the integral in terms of \(u\). Convert \(x\)-expression to \(u\)-expression.
Step 4 Integrate in terms of \(u\). Use standard integration rules.
Step 5 Substitute back the original expression. Return answer in terms of \(x\).
Example idea: In \(\int 2x(x^2+1)^5\,dx\), let \(u=x^2+1\), because \(du=2x\,dx\).

Integration by Parts

Integration by parts is used when the integrand is a product of two functions. It is based on the product rule of differentiation.

Formula
\[ \int u\,dv = uv-\int v\,du \]

This is generally used when the integral contains a product such as \(x e^x\), \(x\sin x\), \(x\cos x\), or \(x\log x\).

LIATE Rule

Choose \(u\) preferably in this order:

  1. Logarithmic
  2. Inverse trigonometric
  3. Algebraic
  4. Trigonometric
  5. Exponential
Exam tip: Use substitution when a function and its derivative appear together. Use integration by parts when two different types of functions are multiplied.

Definite Integrals

A definite integral has lower and upper limits. It gives a numerical value and is often interpreted as the net area under a curve.

Concept Formula Meaning
Definite Integral \(\int_a^b f(x)\,dx\) Integral from \(x=a\) to \(x=b\).
Evaluation \(\int_a^b f(x)\,dx=F(b)-F(a)\) Use antiderivative \(F(x)\).
Zero Interval \(\int_a^a f(x)\,dx=0\) No interval length.
Reversing Limits \(\int_a^b f(x)\,dx=-\int_b^a f(x)\,dx\) Changing order changes sign.
Additive Property \(\int_a^b f(x)\,dx=\int_a^c f(x)\,dx+\int_c^b f(x)\,dx\) Interval can be split.
Important: Definite integrals do not include \(C\), because the constant cancels.

Area of Plane Regions Bounded by Curves

Definite integrals are used to find areas of plane regions bounded by curves, axes and straight lines. The area is always taken as a positive quantity.

Situation Area Formula Use
Area under \(y=f(x)\) above x-axis from \(a\) to \(b\) \(\int_a^b f(x)\,dx\) When \(f(x)\geq 0\).
Area between curve and x-axis \(\int_a^b |f(x)|\,dx\) Use absolute value if curve goes below x-axis.
Area between two curves \(y=f(x)\) and \(y=g(x)\) \(\int_a^b [f(x)-g(x)]\,dx\) Upper curve minus lower curve.
Area with respect to y-axis \(\int_c^d [x_{\text{right}}-x_{\text{left}}]\,dy\) Used when curves are better expressed as functions of \(y\).
Exam tip: Area is always positive. If the definite integral gives a negative value, take the magnitude for geometrical area.

Differential Equations: Order and Degree

A differential equation is an equation involving an unknown function and its derivatives. The order and degree help classify the differential equation.

Concept Meaning Example
Differential Equation An equation containing derivatives. \(\frac{dy}{dx}+y=x\)
Order Highest order derivative present. In \(\frac{d^2y}{dx^2}+y=0\), order is 2.
Degree Power of highest order derivative after making the equation polynomial in derivatives. In \(\left(\frac{dy}{dx}\right)^3+y=0\), degree is 3.
General Solution Solution containing arbitrary constants. \(y=Ce^x\)
Particular Solution Solution after assigning values to arbitrary constants using given conditions. If \(C=2\), then \(y=2e^x\).
Important: Degree is defined only when the differential equation is polynomial in derivatives.

Formation of Differential Equations

A differential equation can be formed from a relation containing arbitrary constants by differentiating and eliminating those constants.

Step Action Example
Step 1 Start with relation containing constant. \(y=Ce^x\)
Step 2 Differentiate with respect to \(x\). \(\frac{dy}{dx}=Ce^x\)
Step 3 Eliminate the arbitrary constant. Since \(y=Ce^x\), \(\frac{dy}{dx}=y\)
Step 4 Write the differential equation. \(\frac{dy}{dx}-y=0\)
Rule: If the original relation has one arbitrary constant, differentiate once. If it has two arbitrary constants, differentiate twice.

First Order and First Degree Differential Equations

A first order and first degree differential equation contains only the first derivative and the first derivative occurs to the first power. These are the most common elementary differential equations.

Type Standard Form Method
Variable Separable \(\frac{dy}{dx}=f(x)g(y)\) Separate variables and integrate.
Linear Differential Equation \(\frac{dy}{dx}+Py=Q\) Use integrating factor.
Homogeneous Type \(\frac{dy}{dx}=F\left(\frac{y}{x}\right)\) Put \(y=vx\).
Growth and Decay \(\frac{dy}{dt}=ky\) Solution is \(y=Ce^{kt}\).
Exam tip: First identify the type of differential equation, then choose the solving method.
Variable Separable Equations

If the equation can be written with all \(y\)-terms on one side and all \(x\)-terms on the other side, it is called variable separable.

\[ \frac{dy}{dx}=f(x)g(y) \] \[ \frac{dy}{g(y)}=f(x)\,dx \] \[ \int \frac{dy}{g(y)}=\int f(x)\,dx \]

Example: \(\frac{dy}{dx}=xy\) becomes \(\frac{dy}{y}=x\,dx\).

Linear Differential Equations

A first order linear differential equation has the form:

\[ \frac{dy}{dx}+Py=Q \] \[ I.F.=e^{\int P\,dx} \] \[ y(I.F.)=\int Q(I.F.)\,dx+C \]

Here, \(I.F.\) means integrating factor.

Applications: Growth and Decay

Growth and decay problems are based on the idea that the rate of change of a quantity is proportional to the quantity itself.

Application Differential Equation Solution Type
Exponential Growth \(\frac{dP}{dt}=kP,\ k>0\) \(P=Ce^{kt}\)
Exponential Decay \(\frac{dP}{dt}=-kP,\ k>0\) \(P=Ce^{-kt}\)
Radioactive Decay \(\frac{dN}{dt}=-kN\) Amount decreases exponentially.
Population Growth \(\frac{dP}{dt}=kP\) Population increases exponentially in simple model.
Remember: Positive \(k\) indicates growth. A negative sign indicates decay.

Solved Examples

Question Method Answer
Evaluate \(\int x^2\,dx\). Use power rule: \[ \int x^2\,dx=\frac{x^3}{3}+C \] \(\frac{x^3}{3}+C\)
Evaluate \(\int 2x(x^2+1)^3\,dx\). Let \(u=x^2+1\). Then \(du=2x\,dx\). \[ \int 2x(x^2+1)^3\,dx=\int u^3\,du \] \[ =\frac{u^4}{4}+C \] \(\frac{(x^2+1)^4}{4}+C\)
Evaluate \(\int x e^x\,dx\). Use integration by parts. Let \(u=x\), \(dv=e^x dx\). Then \(du=dx\), \(v=e^x\). \[ \int x e^x dx = xe^x-\int e^x dx \] \(xe^x-e^x+C\)
Evaluate \(\int_0^2 x\,dx\). \[ \int_0^2 x\,dx=\left[\frac{x^2}{2}\right]_0^2=2 \] 2
Find the area under \(y=x\) from \(x=0\) to \(x=3\). \[ Area=\int_0^3 x\,dx=\left[\frac{x^2}{2}\right]_0^3 \] \(\frac{9}{2}\) square units
Find order and degree of \(\left(\frac{d^2y}{dx^2}\right)^3+\frac{dy}{dx}+y=0\). Highest derivative is \(\frac{d^2y}{dx^2}\), so order is 2. Its power is 3, so degree is 3. Order 2, Degree 3
Solve \(\frac{dy}{dx}=xy\). Separate variables: \[ \frac{dy}{y}=x\,dx \] Integrate: \[ \log |y|=\frac{x^2}{2}+C \] \(y=Ce^{x^2/2}\)
Find the solution of \(\frac{dy}{dt}=ky\). Separate variables: \[ \frac{dy}{y}=k\,dt \] Integrate: \[ \log |y|=kt+C \] \(y=Ce^{kt}\)

Note: In integration, check whether the question needs indefinite integral, definite integral, area, or differential equation solution.

Common Traps and Shortcuts

Common Traps
  • Forgetting \(+C\) in indefinite integrals.
  • Adding \(+C\) unnecessarily in definite integrals.
  • Using wrong substitution in composite expressions.
  • Choosing wrong \(u\) in integration by parts.
  • Forgetting that geometrical area cannot be negative.
  • Confusing order and degree of a differential equation.
  • Finding degree when the equation is not polynomial in derivatives.
  • Not applying initial conditions for particular solutions.
  • Confusing growth model with decay model.
Useful Shortcuts
  • For powers, use \(\int x^n dx=\frac{x^{n+1}}{n+1}+C\).
  • For composite functions, try substitution.
  • For product of unlike functions, try integration by parts.
  • For definite integrals, use upper value minus lower value.
  • For area between curves, use upper curve minus lower curve.
  • For separable differential equations, keep \(y\)-terms and \(x\)-terms separate.
  • For \(\frac{dy}{dx}+Py=Q\), use integrating factor.
  • For growth and decay, remember \(y=Ce^{kt}\).
Exam approach: Identify whether the problem is based on standard integration, substitution, integration by parts, definite integrals, area under curves, order and degree, formation of differential equation, first order differential equation, or growth and decay.

Practice

A) Multiple Choice Questions
  1. \(\int x^3\,dx\) is:
    \(\frac{x^4}{4}+C\) \(3x^2+C\) \(x^4+C\) \(\frac{x^3}{3}+C\)
  2. \(\int \cos x\,dx\) is:
    \(\sin x+C\) \(-\sin x+C\) \(\cos x+C\) \(-\cos x+C\)
  3. \(\int_0^1 1\,dx\) is:
    0 1 2 Not defined
  4. The order of \(\frac{d^3y}{dx^3}+\frac{dy}{dx}+y=0\) is:
    1 2 3 0
  5. The solution form of \(\frac{dy}{dt}=ky\) is:
    \(y=C+kt\) \(y=Ce^{kt}\) \(y=kx+C\) \(y=Ct^k\)
B) Solve the Higher-Order Problems
  1. Evaluate \(\int (3x^2+2x+1)\,dx\). (Hint: Integrate term by term.)
  2. Evaluate \(\int 4x(2x^2+1)^3\,dx\). (Hint: Use substitution \(u=2x^2+1\).)
  3. Evaluate \(\int_1^3 x\,dx\). (Hint: Use \(\left[\frac{x^2}{2}\right]_1^3\).)
  4. Find the order and degree of \(\left(\frac{d^2y}{dx^2}\right)^2+\left(\frac{dy}{dx}\right)^3+y=0\). (Hint: Check highest order derivative and its power.)
  5. Solve \(\frac{dy}{dx}=3y\). (Hint: Separate variables.)
C) Match the Concept with the Correct Rule
Concept Correct Rule / Meaning
Indefinite Integral \(\int f(x)\,dx=F(x)+C\)
Definite Integral \(\int_a^b f(x)\,dx=F(b)-F(a)\)
Integration by Parts \(\int u\,dv=uv-\int v\,du\)
Order of Differential Equation Highest order derivative present
Degree of Differential Equation Power of highest order derivative when polynomial in derivatives
General Solution Solution containing arbitrary constants
Particular Solution Solution obtained after applying given conditions
Growth and Decay Modelled by \(\frac{dy}{dt}=ky\) or \(\frac{dy}{dt}=-ky\)
Integral Calculus and Differential Equations Reminder

Integral Calculus helps in finding antiderivatives, evaluating accumulated quantities, and calculating areas under or between curves. Differential Equations help represent and solve problems involving rates of change. Together, they form a powerful mathematical tool for studying growth, decay, motion, area, and real-world change.

Task: Create five questions using one question each from standard integrals, substitution, integration by parts, definite integrals, and first order differential equations.

Show Suggested Answers
Multiple Choice
  1. \(\frac{x^4}{4}+C\)
    By power rule, \(\int x^3 dx=\frac{x^4}{4}+C\).
  2. \(\sin x+C\)
    \(\int \cos x dx=\sin x+C\).
  3. 1
    \(\int_0^1 1 dx=[x]_0^1=1\).
  4. 3
    Highest order derivative is \(\frac{d^3y}{dx^3}\), so order is 3.
  5. \(y=Ce^{kt}\)
    This is the standard solution of \(\frac{dy}{dt}=ky\).
Higher-Order Problems
  1. \[ \int (3x^2+2x+1)\,dx=x^3+x^2+x+C \] Answer = \(x^3+x^2+x+C\).
  2. Let \(u=2x^2+1\). Then \(du=4x\,dx\). \[ \int 4x(2x^2+1)^3 dx=\int u^3du=\frac{u^4}{4}+C \] Answer = \(\frac{(2x^2+1)^4}{4}+C\).
  3. \[ \int_1^3 x dx=\left[\frac{x^2}{2}\right]_1^3 = \frac{9}{2}-\frac{1}{2}=4 \] Answer = 4.
  4. Highest derivative is \(\frac{d^2y}{dx^2}\), so order is 2. Its power is 2, so degree is 2.
    Answer = Order 2, Degree 2.
  5. \[ \frac{dy}{dx}=3y \] \[ \frac{dy}{y}=3dx \] \[ \log |y|=3x+C \] Answer = \(y=Ce^{3x}\).
Concept Matching
  1. Indefinite Integral → \(\int f(x)\,dx=F(x)+C\)
  2. Definite Integral → \(\int_a^b f(x)\,dx=F(b)-F(a)\)
  3. Integration by Parts → \(\int u\,dv=uv-\int v\,du\)
  4. Order of Differential Equation → Highest order derivative present
  5. Degree of Differential Equation → Power of highest order derivative when polynomial in derivatives
  6. General Solution → Solution containing arbitrary constants
  7. Particular Solution → Solution obtained after applying given conditions
  8. Growth and Decay → Modelled by \(\frac{dy}{dt}=ky\) or \(\frac{dy}{dt}=-ky\)
Clue Explanation

Integration problems are solved by identifying whether the integral is standard, substitution-based, parts-based, or definite. Differential equations are solved by identifying order, degree, type, and whether variables can be separated or an integrating factor is needed.

Exam tips
  • Add \(+C\) only for indefinite integrals.
  • Do not add \(+C\) in definite integrals.
  • Use substitution when a function and its derivative appear together.
  • Use integration by parts for products of functions.
  • For area, identify upper and lower curves.
  • Order means highest derivative order.
  • Degree means power of highest order derivative.
  • Separate variables whenever possible.
  • Use \(y=Ce^{kt}\) for simple growth and decay models.
Practice MCQs