Answer to Question #234229 in Calculus for Sayem

Question

If f is continuous and $\intop f (x) dx=6 (upper limit=2, lower limit=0)$ ,evaluate

$\intop f(2sin\theta) cos\theta d\theta (upper limit=\pi/2 ,lower limit=0)$

Answer 1

Solution:

Given,$\int_0^2 f(x)dx=6$ ...(i)

Let $I=\int_0^{\pi/2} f(2\sin \theta)\cos \theta d\theta$

Put $2\sin \theta=t$

$\Rightarrow 2\cos \theta=\dfrac{dt}{d\theta}$

When $\theta=0,t=0; \theta=\pi/2,t=2$

So, $I=\frac12\int_0^{2}f(t)dt$

$=\frac12\times 6$ [Using (i)]

$=3$