Evaluate $\int_{0}^{3 / 2}|x \sin \pi x| d x$

$\frac{2}{\pi}+\frac{1}{\pi^{2}}$

CBSE Class 12 · 2017 · Definite IntegralsLong Answer

Evaluate

\int_{0}^{3/2} ∣ x sin π x ∣ d x

Answer: $\frac{2}{π} + \frac{1}{π ^{2}}$

[0, 1]

sin π x \geq 0

∣ x sin π x ∣ = x sin π x

; on

[1, \frac{3}{2}]

sin π x \leq 0

∣ x sin π x ∣ = - x sin π x

.
By parts,

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

; call this

F (x)

\int_{0}^{1} x sin π x d x = F (1) - F (0) = \frac{1}{π}

\int_{1}^{3/2} x sin π x d x = F (\frac{3}{2}) - F (1) = - \frac{1}{π ^{2}} - \frac{1}{π}

I = \frac{1}{π} - (- \frac{1}{π ^{2}} - \frac{1}{π}) = \frac{2}{π} + \frac{1}{π ^{2}}

CBSE Class 12 Board · 2017 · Subjective