What is the integral of sin^3(x)cos(cos(x)) ?

The integral of sin^3(x)cos(cos(x)) is -(2(sin(cos(x))-cos(cos(x))cos(x))+sin^2(x)sin(cos(x)))+C

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integral of sin^3(x)cos(cos(x))

\int sin^{3} (x) cos (cos (x)) d x

- (2 (sin (cos (x)) - cos (cos (x)) cos (x)) + sin^{2} (x) sin (cos (x))) + C

Solution steps

\int sin^{3} (x) cos (cos (x)) d x

Apply u-substitution

: \int - cos (u) (1 - u^{2}) d u

= \int - cos (u) (1 - u^{2}) d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int cos (u) (1 - u^{2}) d u

Apply Integration By Parts

: sin (u) (1 - u^{2}) - \int - 2 u sin (u) d u

= - (sin (u) (1 - u^{2}) - \int - 2 u sin (u) d u)

\int - 2 u sin (u) d u = - 2 (- u cos (u) + sin (u))

= - (sin (u) (1 - u^{2}) - (- 2 (- u cos (u) + sin (u))))

Substitute back

u = cos (x)

= - (sin (cos (x)) (1 - cos^{2} (x)) - (- 2 (- cos (x) cos (cos (x)) + sin (cos (x)))))

Simplify

- (sin (cos (x)) (1 - cos^{2} (x)) - (- 2 (- cos (x) cos (cos (x)) + sin (cos (x))))) : - (2 (sin (cos (x)) - cos (cos (x)) cos (x)) + sin^{2} (x) sin (cos (x)))

= - (2 (sin (cos (x)) - cos (cos (x)) cos (x)) + sin^{2} (x) sin (cos (x)))

Add a constant to the solution

= - (2 (sin (cos (x)) - cos (cos (x)) cos (x)) + sin^{2} (x) sin (cos (x))) + C

\int 3 sin^{5} (x) cos^{4} (x) d x

\int cos (\frac{1}{5}) x d x

\int \frac{1}{( x ^{2} + 1 ) \cdot ( x ^{2} + 4 )} d x

\int 3 d^{4} ddd

\int \frac{3 + sin ( 4 x )}{sin ( 4 x )} d x

What is the integral of sin^3(x)cos(cos(x)) ?
The integral of sin^3(x)cos(cos(x)) is -(2(sin(cos(x))-cos(cos(x))cos(x))+sin^2(x)sin(cos(x)))+C