The integral $\int_{-1}^{\frac{3}{2}}\left(\left|\pi^{2} \mathrm{x} \sin (\pi \mathrm{x})\right|\right) \mathrm{dx}$ is equal to

JEE Main · 2025 · 08 Apr 2025 · Shift 2 · Definite Integrals

The integral

\int_{- 1}^{\frac{3}{2}} (π^{2} x sin (π x)) dx

is equal to

Answer: (C)

Write

I = π^{2} \int_{- 1}^{3/2} ∣ x sin π x ∣ d x

. The product

x sin π x

is positive on

(- 1, 0)

and on

(0, 1)

, and negative on

(1, \frac{3}{2})

, so

I = π^{2} [\int_{- 1}^{0} x sin π x d x + \int_{0}^{1} x sin π x d x - \int_{1}^{3/2} x sin π x d x]

. With

\int x sin π x d x = - \frac{x cos π x}{π} + \frac{sin π x}{π ^{2}}

, the three pieces equal

\frac{1}{π}

\frac{1}{π}

and

- \frac{1}{π ^{2}} - \frac{1}{π}

. Hence

I = π^{2} [\frac{1}{π} + \frac{1}{π} + \frac{1}{π ^{2}} + \frac{1}{π}] = π^{2} [\frac{3}{π} + \frac{1}{π ^{2}}] = 3 π + 1

JEE Main · 2025 · 08 Apr 2025 · Shift 2 · Single correct option

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