∫₀π/2 cos xesin xdx = ____.

NCERT Exemplar · Q48

\int_{0}^{\pi/2} \cos x\,e^{\sin x}\,dx =

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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 substitution $u = \sin x$ , which transforms the integral into a simple exponential form. The value is $\boxed{e - 1}$ .

Why this approach works

When you see an integral like $\int \cos x \, e^{\sin x} \, dx$ , your first instinct should be to look for a function and its derivative. Here, $\sin x$ appears inside the exponential, and its derivative $\cos x$ sits right next to it as a factor. That’s the classic signal for the reverse chain rule — or, more formally, integration by substitution.

The power rule for integration ( $\int u^n \, du = \frac{u^{n+1}}{n+1} + C$ ) doesn’t apply directly because we have an exponential, not a power. But the substitution method works the same way: we let $u$ be the “inner” function, rewrite everything in terms of $u$ , and then integrate using the rule for exponentials ( $\int e^u \, du = e^u + C$ ).

Let’s walk through it step by step.

Choose the substitution.

Let $u = \sin x$ . Then $du = \cos x \, dx$ . This is perfect because the integrand has $\cos x \, dx$ multiplied by $e^{\sin x}$ .
Change the limits of integration.

When $x = 0$ , $u = \sin 0 = 0$ .

When $x = \frac{\pi}{2}$ , $u = \sin \frac{\pi}{2} = 1$ .

So the integral becomes:

\int_{0}^{\pi/2} \cos x \, e^{\sin x} \, dx = \int_{0}^{1} e^u \, du

Integrate with respect to $u$ . The antiderivative of $e^u$ is $e^u$ itself. So: $\int_{0}^{1} e^u \, du = \left[ e^u \right]_{0}^{1} = e^{1} - e^{0} = e - 1$ …

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Q. $\int_{0}^{\pi/2} \cos x\,e^{\sin x}\,dx =$ _______.

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 this approach works

Method: Reverse chain rule for $\int g'(x)\,e^{g(x)}\,dx$

Steps

Common Mistakes

Q.∫0π/2cos⁡x esin⁡x dx=\int_{0}^{\pi/2} \cos x\,e^{\sin x}\,dx = ∫0π/2​cosxesinxdx= _______.

Why this approach works

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Q. $\int_{0}^{\pi/2} \cos x\,e^{\sin x}\,dx =$ _______.

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