What is the integral of sin^3(x)cos^{18}(x) ?

The integral of sin^3(x)cos^{18}(x) is -(cos^{19}(x))/(19)+(cos^{21}(x))/(21)+C

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integral of sin^3(x)cos^{18}(x)

\int sin^{3} (x) cos^{18} (x) d x

- \frac{cos ^{19} ( x )}{19} + \frac{cos ^{21} ( x )}{21} + C

Solution steps

\int sin^{3} (x) cos^{18} (x) d x

Rewrite using trig identities

= \int (1 - cos^{2} (x)) sin (x) cos^{18} (x) d x

Apply u-substitution

= \int - u^{18} (1 - u^{2}) d u

Expand

- u^{18} (1 - u^{2}) : - u^{18} + u^{20}

= \int - u^{18} + u^{20} 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^{18} d u + \int u^{20} d u

\int u^{18} d u = \frac{u ^{19}}{19}

\int u^{20} d u = \frac{u ^{21}}{21}

= - \frac{u ^{19}}{19} + \frac{u ^{21}}{21}

Substitute back

u = cos (x)

= - \frac{cos ^{19} ( x )}{19} + \frac{cos ^{21} ( x )}{21}

Add a constant to the solution

= - \frac{cos ^{19} ( x )}{19} + \frac{cos ^{21} ( x )}{21} + C

What is the integral of sin^3(x)cos^{18}(x) ?
The integral of sin^3(x)cos^{18}(x) is -(cos^{19}(x))/(19)+(cos^{21}(x))/(21)+C