The value of the integral integral from -pi/4 to pi/4 of (32cos^4 x / (1+e^sin x)) dx is:

The correct answer is Option 2. Full mathematical solution steps and hints are accessible in the interactive StemCET workspace.

Maths Q18 – JEE-MAIN 2026

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Home Maths IntegralsQuestion #211

The value of the integral

\int_{- \frac{π}{4}}^{\frac{π}{4}} (\frac{32 \cos^{4} x}{1 + e^{\sin x}}) d x

is:

Clue 1 of 3: Property of Definite Integrals

For an integral of the form $\int_{- a}^{a} f (x) d x$ , consider using the property $\int_{- a}^{a} f (x) d x = \int_{0}^{a} [f (x) + f (- x)] d x$ .

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Step 1: Apply Definite Integral Property✦ Active

Let the given integral be $I$ . We use the property $\int_{- a}^{a} f (x) d x = \int_{0}^{a} [f (x) + f (- x)] d x$ . Here, $a = \frac{π}{4}$ and $f (x) = \frac{32 \cos^{4} x}{1 + e^{\sin x}}$ . We find $f (- x)$ :

f (- x) = \frac{32 \cos^{4} (- x)}{1 + e^{\sin (- x)}} = \frac{32 \cos^{4} x}{1 + e^{- \sin x}} = \frac{32 \cos^{4} x}{1 + \frac{1}{e^{\sin x}}} = \frac{32 \cos^{4} x \cdot e^{\sin x}}{e^{\sin x} + 1}

Now, sum $f (x)$ and $f (- x)$ :

f (x) + f (- x) = \frac{32 \cos^{4} x}{1 + e^{\sin x}} + \frac{32 \cos^{4} x \cdot e^{\sin x}}{1 + e^{\sin x}} = \frac{32 \cos^{4} x (1 + e^{\sin x})}{1 + e^{\sin x}} = 32 \cos^{4} x

So the integral simplifies to:

I = \int_{0}^{\frac{π}{4}} 32 \cos^{4} x d x

Step 2: Simplify the Integrand using Trigonometric Identities○ Expand

We use the power reduction formula $\cos^{2} θ = \frac{1 + \cos (2 θ)}{2}$ to simplify $\cos^{4} x$ :

\cos^{4} x = (\cos^{2} x)^{2} = {(\frac{1 + \cos (2 x)}{2})}^{2} = \frac{1 + 2 \cos (2 x) + \cos^{2} (2 x)}{4}

Apply the formula again for $\cos^{2} (2 x)$ :

\cos^{2} (2 x) = \frac{1 + \cos (4 x)}{2}

Substitute back into the expression for $\cos^{4} x$ :

\cos^{4} x = \frac{1 + 2 \cos (2 x) + \frac{1 + \cos (4 x)}{2}}{4} = \frac{2 + 4 \cos (2 x) + 1 + \cos (4 x)}{8} = \frac{3 + 4 \cos (2 x) + \cos (4 x)}{8}

💡 Teacher's Secret Hint

Remember to apply power reduction formulas carefully to avoid errors in coefficients.

Step 3: Evaluate the Definite Integral○ Expand

Substitute the simplified $\cos^{4} x$ back into the integral:

I = 32 \int_{0}^{\frac{π}{4}} \frac{3 + 4 \cos (2 x) + \cos (4 x)}{8} d x = 4 \int_{0}^{\frac{π}{4}} (3 + 4 \cos (2 x) + \cos (4 x)) d x

Integrate term by term:

I = 4 {[3 x + 4 \frac{\sin (2 x)}{2} + \frac{\sin (4 x)}{4}]}_{0}^{\frac{π}{4}} = 4 {[3 x + 2 \sin (2 x) + \frac{1}{4} \sin (4 x)]}_{0}^{\frac{π}{4}}

Evaluate at the limits:

I = 4 [(3 (\frac{π}{4}) + 2 \sin (2 \cdot \frac{π}{4}) + \frac{1}{4} \sin (4 \cdot \frac{π}{4})) - (3 (0) + 2 \sin (0) + \frac{1}{4} \sin (0))]

I = 4 [(\frac{3 π}{4} + 2 \sin (\frac{π}{2}) + \frac{1}{4} \sin (π)) - (0)]

I = 4 [\frac{3 π}{4} + 2 (1) + \frac{1}{4} (0)] = 4 [\frac{3 π}{4} + 2] = 3 π + 8

The value of the integral is $3 π + 8$ .

💡 Teacher's Secret Hint

Be careful with the evaluation of trigonometric functions at the limits, especially $\sin (π / 2)$ and $\sin (π)$ .

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Maths Question 18 – JEE-MAIN 2026

Clue 1 of 3: Property of Definite Integrals

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