What is the integral of 70sin^4(x)cos^3(x) ?

The integral of 70sin^4(x)cos^3(x) is 70((sin^5(x))/5-(sin^7(x))/7)+C

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integral of 70sin^4(x)cos^3(x)

\int 70 sin^{4} (x) cos^{3} (x) d x

70 (\frac{sin ^{5} ( x )}{5} - \frac{sin ^{7} ( x )}{7}) + C

Solution steps

\int 70 sin^{4} (x) cos^{3} (x) d x

Take the constant out:

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

= 70 \cdot \int sin^{4} (x) cos^{3} (x) d x

Rewrite using trig identities

= 70 \cdot \int (1 - sin^{2} (x)) cos (x) sin^{4} (x) d x

Apply u-substitution

= 70 \cdot \int u^{4} (1 - u^{2}) d u

Expand

u^{4} (1 - u^{2}) : u^{4} - u^{6}

= 70 \cdot \int u^{4} - u^{6} 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

= 70 (\int u^{4} d u - \int u^{6} d u)

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

\int u^{6} d u = \frac{u ^{7}}{7}

= 70 (\frac{u ^{5}}{5} - \frac{u ^{7}}{7})

Substitute back

u = sin (x)

= 70 (\frac{sin ^{5} ( x )}{5} - \frac{sin ^{7} ( x )}{7})

Add a constant to the solution

= 70 (\frac{sin ^{5} ( x )}{5} - \frac{sin ^{7} ( x )}{7}) + C

What is the integral of 70sin^4(x)cos^3(x) ?
The integral of 70sin^4(x)cos^3(x) is 70((sin^5(x))/5-(sin^7(x))/7)+C