What is the integral of sin^{100}(x)cos^3(x) ?

The integral of sin^{100}(x)cos^3(x) is (sin^{101}(x))/(101)-(sin^{103}(x))/(103)+C

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integral of sin^{100}(x)cos^3(x)

\int sin^{100} (x) cos^{3} (x) d x

\frac{sin ^{101} ( x )}{101} - \frac{sin ^{103} ( x )}{103} + C

Solution steps

\int sin^{100} (x) cos^{3} (x) d x

Rewrite using trig identities

= \int (1 - sin^{2} (x)) cos (x) sin^{100} (x) d x

Apply u-substitution

: \int u^{100} (1 - u^{2}) d u

= \int u^{100} (1 - u^{2}) d u

Expand

u^{100} (1 - u^{2}) : u^{100} - u^{102}

= \int u^{100} - u^{102} 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^{100} d u - \int u^{102} d u

\int u^{100} d u = \frac{u ^{101}}{101}

\int u^{102} d u = \frac{u ^{103}}{103}

= \frac{u ^{101}}{101} - \frac{u ^{103}}{103}

Substitute back

u = sin (x)

= \frac{sin ^{101} ( x )}{101} - \frac{sin ^{103} ( x )}{103}

Add a constant to the solution

= \frac{sin ^{101} ( x )}{101} - \frac{sin ^{103} ( x )}{103} + C

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What is the integral of sin^{100}(x)cos^3(x) ?
The integral of sin^{100}(x)cos^3(x) is (sin^{101}(x))/(101)-(sin^{103}(x))/(103)+C