Integrate with sin cos

Calculus Level 5

$\large\displaystyle \int_{0}^{\frac{\pi}{4}} (\cos 2 \theta)^{\frac{3}{2}} \cos \theta \, d \theta$

Given the integral above equals to $\dfrac{A}{B \sqrt{C}} \pi$ where $A,B,C$ are positive integers with $A,B$ are coprime and $C$ square-free. Find $B-A-C$ .

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The answer is 11.

1 solution

Refaat M. Sayed
Sep 16, 2015

We can solve this integration by
Assume that $\sin{x}=\sqrt{2} \sin{\theta}$ then we get $d\theta = \frac{\cos \left( x\right) }{\sqrt{2} \cos \left( \theta \right) } dx$ Now the initial integration becom $I= \frac1{\sqrt{2}} \int_0^{\pi/2} dx \, \cos^4{x}$ Which is easy to integrate it $I=\frac1{\sqrt{2}} \int_0^{\pi/2} \, \cos^4{x} dx=\frac{3}{16\sqrt{2} } \pi$ Hence $B-A-C=16-3-2= \boxed{11}$

Integrate with sin cos

Try my set

The answer is 11.

1 solution

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