Unit 4: Integration I - Subjective Questions

MTH110 — Remedial Mathematics • Practice Questions with Detailed Answers

20 questions

Explain integration as the inverse process of differentiation. Illustrate with a suitable example.

State the standard integration formulas for $\int x^n\, dx$ , $\int \frac{1}{x}\, dx$ , $\int e^x\, dx$ , $\int \sin x\, dx$ , and $\int \cos x\, dx$ .

Describe the method of integration by substitution. Explain the general steps involved.

Evaluate $\int \frac{2x}{1+x^2}\, dx$ using the method of substitution.

Evaluate $\int \tan x\, dx$ and $\int \cot x\, dx$ using substitution.

Explain how trigonometric identities are used in integration. Evaluate $\int \sin^2 x\, dx$ .

Evaluate $\int \cos^2 x\, dx$ using a suitable trigonometric identity.

Evaluate $\int \sin 3x \cos 2x\, dx$ using product-to-sum trigonometric identities.

List and derive the standard integrals of the form $\int \frac{dx}{x^2 - a^2}$ and $\int \frac{dx}{x^2 + a^2}$ .

Derive the standard integrals $\int \frac{dx}{\sqrt{a^2 - x^2}}$ and $\int \frac{dx}{\sqrt{x^2 + a^2}}$ .

Evaluate $\int \frac{dx}{x^2 + 6x + 13}$ .

Evaluate $\int \frac{dx}{\sqrt{9 - 4x^2}}$ .

Explain the method of integration by partial fractions. Describe the different forms of partial fraction decomposition for a rational function.

Evaluate $\int \frac{x + 3}{(x-1)(x+2)}\, dx$ using partial fractions.

Evaluate $\int \frac{3x - 2}{(x+1)^2(x+3)}\, dx$ using partial fractions.

Distinguish between integration by substitution and integration by partial fractions with examples.

Integration by Substitution:

Used when the integrand contains a function and (a multiple of) its derivative.
We change the variable using $t = g(x)$ to simplify the integral to a standard form.
Applicable to a wide variety of integrands (trigonometric, exponential, algebraic).
Example: $\int 2x\sqrt{x^2+1}\, dx$ ; let $t = x^2+1$ .

Integration by Partial Fractions:

Used specifically for rational functions $\frac{P(x)}{Q(x)}$ where the denominator can be factorized.
We decompose the fraction into simpler fractions before integrating.
Requires the fraction to be proper (else division is done first).
Example: $\int \frac{1}{(x-1)(x+2)}\, dx$ using $\frac{A}{x-1} + \frac{B}{x+2}$ .

Key difference:

Basis	Substitution	Partial Fractions
Applicability	General integrands	Rational functions only
Technique	Change of variable	Fraction decomposition
Prerequisite	Derivative present	Factorable denominator

Evaluate $\int \frac{dx}{\sqrt{x^2 - 16}}$ and state the standard result used.

Evaluate $\int \frac{1}{1 + \cos x}\, dx$ using a trigonometric identity.

Evaluate $\int \frac{x^2}{(x-1)(x-2)(x-3)}\, dx$ using partial fractions.

Evaluate $\int \sin^3 x\, dx$ using trigonometric identities and substitution.

Question 1

Explain integration as the inverse process of differentiation. Illustrate with a suitable example.

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Answer 1

Integration as the inverse of differentiation:

Differentiation finds the rate of change of a function, while integration reverses this process to recover the original function from its derivative.
If $\frac{d}{dx}F(x) = f(x)$ , then $\int f(x)\, dx = F(x) + C$ , where $C$ is the constant of integration.
The function $F(x)$ is called the antiderivative or primitive of $f(x)$ .

Why the constant $C$ ?

Since the derivative of a constant is zero, infinitely many functions differing by a constant share the same derivative. Hence we add an arbitrary constant $C$ .

Example:

We know $\frac{d}{dx}(x^3) = 3x^2$ .

Therefore:
$\int 3x^2\, dx = x^3 + C$

This shows integration undoes the differentiation, giving back the original function up to a constant.

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Question 3

Describe the method of integration by substitution. Explain the general steps involved.

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Answer 3

Integration by substitution:

This method simplifies an integral by changing the variable of integration so that it reduces to a standard form.

General principle:

If $I = \int f(g(x))\, g'(x)\, dx$ , we substitute $t = g(x)$ so that $dt = g'(x)\, dx$ , giving:
$I = \int f(t)\, dt$

Steps involved:

Identify a function whose derivative also appears (or nearly appears) in the integrand.
Let $t$ equal that function, i.e. $t = g(x)$ .
Differentiate to find $dt = g'(x)\, dx$ and replace $dx$ accordingly.
Rewrite the entire integral in terms of $t$ .
Integrate with respect to $t$ .
Substitute back $t = g(x)$ to express the answer in the original variable.

Example:
$\int 2x\, e^{x^2}\, dx$

Let $t = x^2 \Rightarrow dt = 2x\, dx$ . Then:
$\int e^t\, dt = e^t + C = e^{x^2} + C$

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Question 13

Explain the method of integration by partial fractions. Describe the different forms of partial fraction decomposition for a rational function.

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Answer 13

Integration by partial fractions:

A proper rational function $\frac{P(x)}{Q(x)}$ (where degree of $P$ < degree of $Q$ ) can be expressed as a sum of simpler fractions that are easy to integrate.

If the fraction is improper (degree of $P \geq$ degree of $Q$ ), first divide to make it proper.

Forms of partial fraction decomposition:

Non-repeated linear factors:
$\frac{px + q}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}$
Repeated linear factor:
$\frac{px + q}{(x-a)^2} = \frac{A}{x-a} + \frac{B}{(x-a)^2}$
Non-repeated quadratic factor (irreducible):
$\frac{px^2 + qx + r}{(x-a)(x^2 + bx + c)} = \frac{A}{x-a} + \frac{Bx + C}{x^2 + bx + c}$

Steps:

Factorize the denominator.
Write the appropriate partial fraction form with unknown constants.
Determine the constants by equating coefficients or substituting convenient values of $x$ .
Integrate each simple fraction separately.

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Question 16

Distinguish between integration by substitution and integration by partial fractions with examples.

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Answer 16

Integration by Substitution:

Used when the integrand contains a function and (a multiple of) its derivative.
We change the variable using $t = g(x)$ to simplify the integral to a standard form.
Applicable to a wide variety of integrands (trigonometric, exponential, algebraic).
Example: $\int 2x\sqrt{x^2+1}\, dx$ ; let $t = x^2+1$ .

Integration by Partial Fractions:

Used specifically for rational functions $\frac{P(x)}{Q(x)}$ where the denominator can be factorized.
We decompose the fraction into simpler fractions before integrating.
Requires the fraction to be proper (else division is done first).
Example: $\int \frac{1}{(x-1)(x+2)}\, dx$ using $\frac{A}{x-1} + \frac{B}{x+2}$ .

Key difference:

Basis	Substitution	Partial Fractions
Applicability	General integrands	Rational functions only
Technique	Change of variable	Fraction decomposition
Prerequisite	Derivative present	Factorable denominator

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Unit3 Unit5

Unit 4: Integration I - Subjective Questions

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