What is the integral of 2sin^3(x)cos^5(x) ?

The integral of 2sin^3(x)cos^5(x) is 2(-(cos^6(x))/6+(cos^8(x))/8)+C

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integral of 2sin^3(x)cos^5(x)

\int 2 sin^{3} (x) cos^{5} (x) d x

2 (- \frac{cos ^{6} ( x )}{6} + \frac{cos ^{8} ( x )}{8}) + C

Solution steps

\int 2 sin^{3} (x) cos^{5} (x) d x

Take the constant out:

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

= 2 \cdot \int sin^{3} (x) cos^{5} (x) d x

Rewrite using trig identities

= 2 \cdot \int (1 - cos^{2} (x)) sin (x) cos^{5} (x) d x

Apply u-substitution

: \int - u^{5} (1 - u^{2}) d u

= 2 \cdot \int - u^{5} (1 - u^{2}) d u

Expand

- u^{5} (1 - u^{2}) : - u^{5} + u^{7}

= 2 \cdot \int - u^{5} + u^{7} 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

= 2 (- \int u^{5} d u + \int u^{7} d u)

\int u^{5} d u = \frac{u ^{6}}{6}

\int u^{7} d u = \frac{u ^{8}}{8}

= 2 (- \frac{u ^{6}}{6} + \frac{u ^{8}}{8})

Substitute back

u = cos (x)

= 2 (- \frac{cos ^{6} ( x )}{6} + \frac{cos ^{8} ( x )}{8})

Add a constant to the solution

= 2 (- \frac{cos ^{6} ( x )}{6} + \frac{cos ^{8} ( x )}{8}) + C

\frac{\partial}{\partial y} (x y cos (x y) + sin (x y))

f (x) = arctan (3 x)

\frac{s + 1}{s ^{2} + 4}

\frac{( x ^{2} + 2 ) ^{\frac{3}{2}}}{3}

f^{^{'}} (x) = tan (x) + cot (x)

What is the integral of 2sin^3(x)cos^5(x) ?
The integral of 2sin^3(x)cos^5(x) is 2(-(cos^6(x))/6+(cos^8(x))/8)+C