Trig and trig inverse integrand!?

Calculus Level 4

$\int _{ -1 }^{ 1 } \sin { (\arcsin { x }) } \times \sin { (\arccos { x }) } \times \cos { (\arccos { x }) } \times \cos { (\arcsin { x }) } \, dx$

If the integral above can be expressed in the form $\dfrac { A }{ B }$ , where ${ A }$ and ${ B }$ are positive coprime integers, find $A+B$ .

If you enjoyed the problem, here is part 2 .

This problem is original.

The answer is 19.

1 solution

Rishabh Jain
Jan 8, 2016

$\sin(\sin^{-1}x)=\cos(\cos^{-1}x)=\color{#69047E}{x}$ $\cos(\sin^{-1}x)=\sin(\cos^{-1}x)=\color{#69047E}{\sqrt{1-x^2}}$ Integral simplifies to:
$I=\int _{ -1 }^{ 1 }x^2(1-x^2)=\frac{2}{3}-\frac{2}{5}=\color{#D61F06}{\frac{4}{15}}$ Hence, 15+4=19.

Trig and trig inverse integrand!?

If you enjoyed the problem, here is part 2 .

This problem is original.

The answer is 19.

1 solution

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