Partial Sine

Calculus Level 3

$\large \int_{\pi/4}^{3\pi/4} \dfrac x{1+\sin(x)} \, dx$

If the integral above equals to $\pi(\sqrt A - \sqrt B)$ for positive integers $A$ and $B$ , find the value of $A+B$ .

The answer is 3.

1 solution

Iqbal Choudhury
Sep 6, 2015

Okay { denotes the integral symbol(long s) and the angles are 135* & 45* denoted by a and b respectively.
Now let I = { x/1+sinx
. I= { pi-x/1+sin(pi-x)
. adding we get
2I = pi {1/1+sinx
;.2I/pi = { 1-sinx/1-(sinx)^2
= { (secx)^2- secx.tanx
; integrating we get
. = tanx| - secx|
. putiing the limits we get
2I = pi 2(√2-1);
I= pi(√2-1)
I.E A=2 and B =1
Therefore A+B=3

Partial Sine

The answer is 3.

1 solution

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