Question Statement

Evaluate the indefinite integral:

$\int sec^{2} (5 x - 1) d x$

Background and Explanation

This problem requires the method of substitution (u-substitution) to integrate a composite function where the inner function is linear ( $5 x - 1$ ). You need to recall the standard integral formula for the secant squared function and how to adjust for the chain rule factor.

Solution

Let $I$ denote the integral:

$I = \int sec^{2} (5 x - 1) d x$

Setting up the Substitution

To handle the argument $(5 x - 1)$ , we use substitution. Let:

$t = 5 x - 1$

Differentiating both sides with respect to $x$ :

$\frac{d t}{d x} = 5 \Rightarrow d t = 5 d x$

Solving for $d x$ :

$d x = \frac{1}{5} d t$

Transforming the Integral

Substitute $t = 5 x - 1$ and $d x = \frac{1}{5} d t$ into the original integral:

$I = \int sec^{2} (t) \cdot \frac{1}{5} d t$

Factor out the constant $\frac{1}{5}$ :

$I = \frac{1}{5} \int sec^{2} t d t$

Evaluating the Integral

Using the standard integral formula $\int sec^{2} x d x = tan x + C$ :

$I = \frac{1}{5} tan (t) + C$

Back-Substitution

Replace $t$ with the original expression $5 x - 1$ :

$I = \frac{1}{5} tan (5 x - 1) + C$

Verification by Differentiation

To verify, differentiate the result with respect to $x$ :

$\frac{d}{d x} [\frac{1}{5} tan (5 x - 1) + C]$

Apply the constant multiple rule and chain rule:

$= \frac{1}{5} \cdot sec^{2} (5 x - 1) \cdot \frac{d}{d x} (5 x - 1) + 0$

Compute the derivative of the inner function:

$= \frac{1}{5} \cdot sec^{2} (5 x - 1) \cdot 5$

Simplify:

$= sec^{2} (5 x - 1)$

Since the derivative equals the original integrand, the solution is correct.

Key Formulas or Methods Used

Integration by Substitution: $\int f (g (x)) \cdot g^{'} (x) d x = \int f (u) d u$ where $u = g (x)$
Standard Integral: $\int sec^{2} x d x = tan x + C$
Chain Rule: $\frac{d}{d x} [f (g (x))] = f^{'} (g (x)) \cdot g^{'} (x)$
Derivative of Tangent: $\frac{d}{d x} [tan x] = sec^{2} x$

Summary of Steps

Substitute: Let $t = 5 x - 1$ to simplify the composite argument
Compute Differential: Find $d t = 5 d x$ , therefore $d x = \frac{d t}{5}$
Rewrite Integral: Transform to $\frac{1}{5} \int sec^{2} t d t$
Integrate: Apply $\int sec^{2} t d t = tan t + C$ to obtain $\frac{1}{5} tan t + C$
Back-Substitute: Replace $t$ with $5 x - 1$ to get $\frac{1}{5} tan (5 x - 1) + C$
Verify: Differentiate using the chain rule to confirm the result equals $sec^{2} (5 x - 1)$

Verification by Differentiation

To verify, differentiate the result with respect to

x

\frac{d}{d x} [\frac{1}{5} tan (5 x - 1) + C]

Apply the constant multiple rule and chain rule:

= \frac{1}{5} \cdot sec^{2} (5 x - 1) \cdot \frac{d}{d x} (5 x - 1) + 0

Compute the derivative of the inner function:

= \frac{1}{5} \cdot sec^{2} (5 x - 1) \cdot 5

Simplify:

= sec^{2} (5 x - 1)

Since the derivative equals the original integrand, the solution is correct.

Summary of Steps

Substitute: Let

t = 5 x - 1

to simplify the composite argument

Compute Differential: Find

d t = 5 d x

, therefore

d x = \frac{d t}{5}

Rewrite Integral: Transform to

\frac{1}{5} \int sec^{2} t d t

Integrate: Apply

\int sec^{2} t d t = tan t + C

to obtain

\frac{1}{5} tan t + C

Back-Substitute: Replace

t

with

5 x - 1

to get

\frac{1}{5} tan (5 x - 1) + C

Verify: Differentiate using the chain rule to confirm the result equals

sec^{2} (5 x - 1)

3.2 Q-8

Question Statement

Background and Explanation

Solution

Setting up the Substitution

Transforming the Integral

Evaluating the Integral

Back-Substitution

Verification by Differentiation

Key Formulas or Methods Used

Summary of Steps

3.2 Q-8

Question Statement

Background and Explanation

Solution

Setting up the Substitution

Transforming the Integral

Evaluating the Integral

Back-Substitution

Verification by Differentiation

Key Formulas or Methods Used

Summary of Steps