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

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

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integral of sin^2(t)cos^3(t)

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

\frac{sin ^{3} ( t )}{3} - \frac{sin ^{5} ( t )}{5} + C

Solution steps

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

Rewrite using trig identities

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

Apply u-substitution

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

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

Expand

u^{2} (1 - u^{2}) : u^{2} - u^{4}

= \int u^{2} - u^{4} 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^{2} d u - \int u^{4} d u

\int u^{2} d u = \frac{u ^{3}}{3}

\int u^{4} d u = \frac{u ^{5}}{5}

= \frac{u ^{3}}{3} - \frac{u ^{5}}{5}

Substitute back

u = sin (t)

= \frac{sin ^{3} ( t )}{3} - \frac{sin ^{5} ( t )}{5}

Add a constant to the solution

= \frac{sin ^{3} ( t )}{3} - \frac{sin ^{5} ( t )}{5} + C

\frac{d}{d x} (\frac{( 1 - x )}{2 x})

y^{^{'}} = 0.19 y + 200

\int 5 e^{4} d x

y = x^{\frac{4}{7}} + 6 x

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