What is the integral of 12sin(x)cos^3(x) ?

The integral of 12sin(x)cos^3(x) is -3cos^4(x)+C

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integral of 12sin(x)cos^3(x)

\int 12 sin (x) cos^{3} (x) d x

- 3 cos^{4} (x) + C

Solution steps

\int 12 sin (x) cos^{3} (x) d x

Take the constant out:

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

= 12 \cdot \int sin (x) cos^{3} (x) d x

Apply u-substitution

: \int - u^{3} d u

= 12 \cdot \int - u^{3} d u

Take the constant out:

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

= 12 (- \int u^{3} d u)

Apply the Power Rule

: \frac{u ^{4}}{4}

= 12 (- \frac{u ^{4}}{4})

Substitute back

u = cos (x)

= 12 (- \frac{cos ^{4} ( x )}{4})

Simplify

12 (- \frac{cos ^{4} ( x )}{4}) : - 3 cos^{4} (x)

= - 3 cos^{4} (x)

Add a constant to the solution

= - 3 cos^{4} (x) + C

\int \frac{2 x + 1}{x ^{2} + 2 x + 3} d x

- 3 x

\int x^{2} - 10 x + 25 d x

\int \frac{1}{( x ^{2} + 2 x + 1 )} d x

\int (\frac{- x}{x ^{2} + 1}) d x

What is the integral of 12sin(x)cos^3(x) ?
The integral of 12sin(x)cos^3(x) is -3cos^4(x)+C