What is the integral of (1-cos^2(x))sin(x)cos^2(x) ?

The integral of (1-cos^2(x))sin(x)cos^2(x) is -1/3 cos^3(x)+1/5 cos^5(x)+C

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integral of (1-cos^2(x))sin(x)cos^2(x)

\int (1 - cos^{2} (x)) sin (x) cos^{2} (x) d x

- \frac{1}{3} cos^{3} (x) + \frac{1}{5} cos^{5} (x) + C

Solution steps

\int (1 - cos^{2} (x)) sin (x) cos^{2} (x) d x

Apply u-substitution

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

Take the constant out:

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

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

Expand

u^{2} (1 - u^{2}) : u^{2} - u^{4}

= - \int u^{2} - u^{4} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - (\int u^{2} d u - \int u^{4} d u)

\int u^{2} d u = \frac{u ^{3}}{3}

\int u^{4} d u = \frac{u ^{5}}{5}

= - (\frac{u ^{3}}{3} - \frac{u ^{5}}{5})

Substitute back

u = cos (x)

= - (\frac{cos ^{3} ( x )}{3} - \frac{cos ^{5} ( x )}{5})

Simplify

- (\frac{cos ^{3} ( x )}{3} - \frac{cos ^{5} ( x )}{5}) : - \frac{1}{3} cos^{3} (x) + \frac{1}{5} cos^{5} (x)

= - \frac{1}{3} cos^{3} (x) + \frac{1}{5} cos^{5} (x)

Add a constant to the solution

= - \frac{1}{3} cos^{3} (x) + \frac{1}{5} cos^{5} (x) + C

What is the integral of (1-cos^2(x))sin(x)cos^2(x) ?
The integral of (1-cos^2(x))sin(x)cos^2(x) is -1/3 cos^3(x)+1/5 cos^5(x)+C