What is the integral of sin^8(x)cos^3(x) ?

The integral of sin^8(x)cos^3(x) is (sin^9(x))/9-(sin^{11}(x))/(11)+C

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integral of sin^8(x)cos^3(x)

\int sin^{8} (x) cos^{3} (x) d x

\frac{sin ^{9} ( x )}{9} - \frac{sin ^{11} ( x )}{11} + C

Solution steps

\int sin^{8} (x) cos^{3} (x) d x

Rewrite using trig identities

= \int (1 - sin^{2} (x)) cos (x) sin^{8} (x) d x

Apply u-substitution

: \int u^{8} (1 - u^{2}) d u

= \int u^{8} (1 - u^{2}) d u

Expand

u^{8} (1 - u^{2}) : u^{8} - u^{10}

= \int u^{8} - u^{10} 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^{8} d u - \int u^{10} d u

\int u^{8} d u = \frac{u ^{9}}{9}

\int u^{10} d u = \frac{u ^{11}}{11}

= \frac{u ^{9}}{9} - \frac{u ^{11}}{11}

Substitute back

u = sin (x)

= \frac{sin ^{9} ( x )}{9} - \frac{sin ^{11} ( x )}{11}

Add a constant to the solution

= \frac{sin ^{9} ( x )}{9} - \frac{sin ^{11} ( x )}{11} + C

\int_{- \infty}^{0} 4^{x} d x

y = x \cdot e^{x}

\int \frac{15}{1 - cos ( 3 x )} d x

\frac{d}{d x} (357 x + 45 (x + \frac{1}{x})^{3})

(t - 1) (3 (t - 1)^{2} + 9 (t - 1) + 3)

What is the integral of sin^8(x)cos^3(x) ?
The integral of sin^8(x)cos^3(x) is (sin^9(x))/9-(sin^{11}(x))/(11)+C