Integration problem

Calculus Level 3

$I=\int_0^\frac \pi 4 \sin \left( x\right) dx +\int_0^\frac \pi 4 \sin ^{2}\left( x\right) dx + \int_0^\frac \pi 4 \sin ^{3}\left( x\right) dx + \int_0^\frac \pi 4\sin ^{4}\left( x\right) dx+\cdots$ Given that $I=\sqrt{A} - \dfrac{\pi}{B}$ , find $A+B$ .

The answer is 6.

1 solution

Rishabh Jain
Jan 29, 2016

$\large I=\int_{0}^\frac{\pi}{4} \sin x +\sin^2 x+ \sin^3 x+...$ Since |sin x| $\leq$ 1, integral is the sum of an infinite G.P whose first term as well as common difference is $\color{#EC7300}{\sin x}$
$\therefore I=\int_{0}^{\frac{\pi}{4}}\dfrac{\sin x}{1-\sin x}$ $=\int_{0}^{\frac{\pi}{4}} \sec x \tan x +\tan^2x$ $=\sec x +\tan x -x |_{\small 0}^{\small{ \frac{\pi}{4}}}$ $=\large \sqrt2-\dfrac{\pi}{4}$ $\Large 4+2=\boxed{\color{#20A900}{\boxed{6}}}$

Integration problem

The answer is 6.

1 solution

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