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

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

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integral of sin^2(x)sin(x)

\int sin^{2} (x) sin (x) d x

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

Solution steps

\int sin^{2} (x) sin (x) d x

Rewrite using trig identities

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

Apply u-substitution

: \int - 1 + u^{2} d u

= \int - 1 + u^{2} 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 1 d u + \int u^{2} d u

\int 1 d u = u

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

= - u + \frac{u ^{3}}{3}

Substitute back

u = cos (x)

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

Add a constant to the solution

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

\int_{1}^{10} \frac{1}{( x - 2 ) ^{\frac{1}{3}}} d x

\frac{\partial}{\partial x} (4 x^{4 y})

\int (- 16 cos (2 t)) d t

\int (\frac{sin ( x )}{cos ( x )}) d x

\frac{d}{d x} (\frac{y - x ^{2} + 2}{x - y})

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