What is the integral of 10sin^3(x)cos^7(x) ?

The integral of 10sin^3(x)cos^7(x) is -5/4 cos^8(x)+cos^{10}(x)+C

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integral of 10sin^3(x)cos^7(x)

\int 10 sin^{3} (x) cos^{7} (x) d x

- \frac{5}{4} cos^{8} (x) + cos^{10} (x) + C

Solution steps

\int 10 sin^{3} (x) cos^{7} (x) d x

Take the constant out:

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

= 10 \cdot \int sin^{3} (x) cos^{7} (x) d x

Rewrite using trig identities

= 10 \cdot \int (1 - cos^{2} (x)) sin (x) cos^{7} (x) d x

Apply u-substitution

= 10 \cdot \int - u^{7} (1 - u^{2}) d u

Expand

- u^{7} (1 - u^{2}) : - u^{7} + u^{9}

= 10 \cdot \int - u^{7} + u^{9} 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

= 10 (- \int u^{7} d u + \int u^{9} d u)

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

\int u^{9} d u = \frac{u ^{10}}{10}

= 10 (- \frac{u ^{8}}{8} + \frac{u ^{10}}{10})

Substitute back

u = cos (x)

= 10 (- \frac{cos ^{8} ( x )}{8} + \frac{cos ^{10} ( x )}{10})

Simplify

10 (- \frac{cos ^{8} ( x )}{8} + \frac{cos ^{10} ( x )}{10}) : - \frac{5}{4} cos^{8} (x) + cos^{10} (x)

= - \frac{5}{4} cos^{8} (x) + cos^{10} (x)

Add a constant to the solution

= - \frac{5}{4} cos^{8} (x) + cos^{10} (x) + C

What is the integral of 10sin^3(x)cos^7(x) ?
The integral of 10sin^3(x)cos^7(x) is -5/4 cos^8(x)+cos^{10}(x)+C