Evaluate $\int_{-a}^{a} \sqrt{\frac{a-x}{a+x}} d x$

CBSE Class 12 · 2008 · Definite IntegralsLong Answer

Evaluate

\int_{- a}^{a} \frac{a - x}{a + x} d x

Answer: $aπ$

Rationalise by multiplying inside the root by

(a - x)

\frac{a - x}{a + x} = \frac{a - x}{a ^{2} - x ^{2}} = \frac{a}{a ^{2} - x ^{2}} - \frac{x}{a ^{2} - x ^{2}}

Over

[- a, a]

the first term is even and the second is odd (its integral vanishes):

I = 2 a \int_{0}^{a} \frac{d x}{a ^{2} - x ^{2}} = 2 a [sin^{- 1} \frac{x}{a}]_{0}^{a} = 2 a \cdot \frac{π}{2} = aπ

CBSE Class 12 Board · 2008 · Subjective