Integrate the following function: $ \sec^2 (7 - 4x) $

Integrate the following function: sec² (7 - 4x) – Class 12 Mathematics

Exercise 7.2 · Q22

\sec^2 (7 - 4x)

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Concept understanding — U Substitution

U Substitution: The Reverse Chain Rule

The chain rule differentiates composite functions: the derivative of $\sin(x^2)$ is $\cos(x^2) \cdot 2x$ — differentiate the outer function, then multiply by the derivative of the inside. Integration asks the reverse: given $\cos(x^2) \cdot 2x$ , find the original function. That's what u substitution does — it reverses the chain rule.

The Core Intuition

When an integral looks like "a function times the derivative of its inside," substitute the inside with $u$ and the derivative of the inside with $du$ . Consider:

\int 2x \cos(x^2) \, dx

Here $2x$ is the derivative of $x^2$ , and $x^2$ is the inside of $\cos(x^2)$ . Let $u = x^2$ , so $du = 2x \, dx$ :

\int \cos(u) \, du = \sin(u) + C = \sin(x^2) + C

Check: the derivative of $\sin(x^2)$ is $\cos(x^2) \cdot 2x$ .

The Precise Statement

\int f(g(x)) \cdot g'(x) \, dx = \int f(u) \, du \quad \text{where } u = g(x), \, du = g'(x) \, dx

Valid provided $g$ is differentiable and the resulting integral in $u$ is simpler.

The Step-by-Step Method

Identify a function $g(x)$ whose derivative $g'(x)$ also appears (possibly up to a constant factor).
Set $u = g(x)$ , compute $du = g'(x) \, dx$ .
Rewrite the entire integral in $u$ and $du$ — every $x$ and $dx$ must be replaced.
Integrate with respect to $u$ .
Substitute back $u = g(x)$ .

Watch out

You cannot mix variables. If any $x$ remains after substitution, you chose the wrong $u$ (or must solve for $x$ in terms of $u$ — rare).

A Second Example (with a constant factor)

Evaluate $\int x \sqrt{x^2 + 1} \, dx$ . Let $u = x^2 + 1$ , so $x \, dx = \frac{1}{2} du$ :

\int \sqrt{u} \cdot \tfrac{1}{2} du = \tfrac{1}{2} \cdot \tfrac{2}{3} u^{3/2} + C = \tfrac{1}{3} (x^2 + 1)^{3/2} + C

When Does It Work?

When the integrand is something times the derivative of something inside. Common patterns:

$x \cdot f(x^2)$ — derivative of $x^2$ is $2x$ , so $u = x^2$
$e^{g(x)} \cdot g'(x)$ — derivative of $g(x)$ appears
$\frac{g'(x)}{g(x)}$ — leads to $\log|g(x)|$ …

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The key idea is to use the substitution $u = 7 - 4x$ , which turns the integral into a standard form. The final answer is $-\frac{1}{4} \tan(7 - 4x) + C$ .

Why U-Substitution Works Here

When you see a function like $\sec^2$ wrapped around a linear expression like $7 - 4x$ , your first instinct should be: the derivative of the inside will simplify things. The derivative of $7 - 4x$ is $-4$ , a constant. That constant can be pulled out of the integral, leaving you with a pure $\sec^2 u$ — which integrates directly to $\tan u$ .

Without substitution, you'd be stuck trying to guess a composite antiderivative. With it, the problem becomes mechanical.

Step-by-Step Solution

Set up the substitution. Let $u = 7 - 4x$ . Then differentiate:

\frac{du}{dx} = -4 \quad \Rightarrow \quad du = -4 \, dx \quad \Rightarrow \quad dx = -\frac{1}{4} \, du.

Rewrite the integral in terms of $u$ . Replace $7 - 4x$ with $u$ , and $dx$ with $-\frac{1}{4} du$ :

\int \sec^2(7 - 4x) \, dx = \int \sec^2(u) \cdot \left(-\frac{1}{4}\right) du = -\frac{1}{4} \int \sec^2 u \, du.

Integrate the standard form. The antiderivative of $\sec^2 u$ is $\tan u$ (this is a fundamental derivative fact: $\frac{d}{du} \tan u = \sec^2 u$ ). So:

-\frac{1}{4} \int \sec^2 u \, du = -\frac{1}{4} \tan u + C.

Substitute back. Replace $u$ with $7 - 4x$ : …

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Q.Integrate the following function: $\sec^2 (7 - 4x)$

Concept understanding — U Substitution

U Substitution: The Reverse Chain Rule

The Core Intuition

The Precise Statement

The Step-by-Step Method

A Second Example (with a constant factor)

When Does It Work?

Why U-Substitution Works Here

Step-by-Step Solution

Method: Integrate a standard trig form with a linear argument

Steps

Common Mistakes

Q.Integrate the following function: sec⁡2(7−4x)\sec^2 (7 - 4x)sec2(7−4x)

Why U-Substitution Works Here

Step-by-Step Solution

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Q.Integrate the following function: $\sec^2 (7 - 4x)$

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