What is the integral of 6sin(x)cos^2(x) ?

The integral of 6sin(x)cos^2(x) is -2cos^3(x)+C

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integral of 6sin(x)cos^2(x)

\int 6 sin (x) cos^{2} (x) d x

- 2 cos^{3} (x) + C

Solution steps

\int 6 sin (x) cos^{2} (x) d x

Take the constant out:

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

= 6 \cdot \int sin (x) cos^{2} (x) d x

Apply u-substitution

: \int - u^{2} d u

= 6 \cdot \int - u^{2} d u

Take the constant out:

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

= 6 (- \int u^{2} d u)

Apply the Power Rule

: \frac{u ^{3}}{3}

= 6 (- \frac{u ^{3}}{3})

Substitute back

u = cos (x)

= 6 (- \frac{cos ^{3} ( x )}{3})

Simplify

6 (- \frac{cos ^{3} ( x )}{3}) : - 2 cos^{3} (x)

= - 2 cos^{3} (x)

Add a constant to the solution

= - 2 cos^{3} (x) + C

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\int 1 + 4 x^{3} (12 x^{2}) d x

What is the integral of 6sin(x)cos^2(x) ?
The integral of 6sin(x)cos^2(x) is -2cos^3(x)+C