Question

$\int _ { 0 } ^ { \pi } x$ f (sin x) dx is equal to

• A. πdx
• B. $\pi \int _ { 0 } ^ { \frac { \pi } { 2 } } f ( \sin x )$ dx
• C. dx
• D. None of these

Answer 1

Answer: (B)

$\pi \int _ { 0 } ^ { \frac { \pi } { 2 } } f ( \sin x )$ dx

Answer 2

I = $\int _ { 0 } ^ { \pi } x$ f (sin x) dx

= $\int _ { 0 } ^ { \pi } ( \pi - x )$ f sin (π – x) dx = π $\int _ { 0 } ^ { \pi } f$(sin x) dx – I

⇒ I = $\frac { \pi } { 2 }$ $\int _ { 0 } ^ { \pi } f$(sin x) dx.

Again, I = $\frac { \pi } { 2 }$ $\int _ { 0 } ^ { \pi } f$(sin x) dx = 2 $\frac { \pi } { 2 }$ $\int _ { 0 } ^ { \pi / 2 } f ( \sin x ) d x$

= π $\int _ { 0 } ^ { \pi / 2 } f ( \sin x ) d x$