If $\int_{0}^{a} \dfrac{dx}{1+4x^2} = \dfrac{\pi}{8}$, then $a = $ _______.

If ∫₀a (dx)/(1+4x²) = (π)/(8), then a = ____.

NCERT Exemplar · Q50

\int_{0}^{a} \dfrac{dx}{1+4x^2} = \dfrac{\pi}{8}

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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 recognise the integrand as the derivative of $\frac{1}{2}\tan^{-1}(2x)$ , then use the given value of the definite integral to solve for $a$ . The answer is $a = \frac{1}{2}$ .

We are given:

\int_{0}^{a} \frac{dx}{1+4x^2} = \frac{\pi}{8}

and need to find $a$ .

The integrand $\frac{1}{1+4x^2}$ looks almost like the derivative of $\tan^{-1} x$ , which is $\frac{1}{1+x^2}$ . The difference is the $4x^2$ instead of $x^2$ . This suggests a substitution that turns $4x^2$ into $u^2$ .

Why substitution works here: If we let $u = 2x$ , then $du = 2\,dx$ , so $dx = \frac{du}{2}$ . Also $4x^2 = (2x)^2 = u^2$ . The integrand becomes $\frac{1}{1+u^2} \cdot \frac{du}{2}$ , which is exactly $\frac{1}{2}$ times the derivative of $\tan^{-1} u$ . This is a standard pattern: whenever you see $\frac{dx}{a^2 + x^2}$ , think of $\frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right)$ .

Let’s work through it step by step.

Set up the substitution.

Let $u = 2x$ . Then $du = 2\,dx$ , so $dx = \frac{du}{2}$ .

When $x = 0$ , $u = 0$ . When $x = a$ , $u = 2a$ .
Rewrite the integral.

\int_{0}^{a} \frac{dx}{1+4x^2} = \int_{u=0}^{u=2a} \frac{1}{1+u^2} \cdot \frac{du}{2} = \frac{1}{2} \int_{0}^{2a} \frac{du}{1+u^2}

Evaluate the standard integral. We know $\int \frac{du}{1+u^2} = \tan^{-1} u + C$ . So:

\frac{1}{2} \int_{0}^{2a} \frac{du}{1+u^2} = \frac{1}{2} \left[ \tan^{-1} u \right]_{0}^{2a} = \frac{1}{2} \left( \tan^{-1}(2a) - \tan^{-1}(0) \right)

Since $\tan^{-1}(0) = 0$ , this simplifies to:

\frac{1}{2} \tan^{-1}(2a)

Set equal to the given value. The problem states this equals $\frac{\pi}{8}$ :

\frac{1}{2} \tan^{-1}(2a) = \frac{\pi}{8}

Solve for $a$ . …

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Q.If $\int_{0}^{a} \dfrac{dx}{1+4x^2} = \dfrac{\pi}{8}$ , then $a =$ _______.

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?

Method: Matching $\int\frac{dx}{1+k^2x^2}$ to the arctangent form

Steps

Common Mistakes

Q.If ∫0adx1+4x2=π8\int_{0}^{a} \dfrac{dx}{1+4x^2} = \dfrac{\pi}{8}∫0a​1+4x2dx​=8π​, then a=a = a= _______.

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Q.If $\int_{0}^{a} \dfrac{dx}{1+4x^2} = \dfrac{\pi}{8}$ , then $a =$ _______.

Unlock everything free for 14 days