What is the integral of (cos(t))/((4sin(t)+12)^2) ?

The integral of (cos(t))/((4sin(t)+12)^2) is -1/(4(4sin(t)+12))+C

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integral of (cos(t))/((4sin(t)+12)^2)

\int \frac{cos ( t )}{( 4 sin ( t ) + 12 ) ^{2}} d t

- \frac{1}{4 ( 4 sin ( t ) + 12 )} + C

Solution steps

\int \frac{cos ( t )}{( 4 sin ( t ) + 12 ) ^{2}} d t

Apply u-substitution

: \int \frac{1}{4 u ^{2}} d u

= \int \frac{1}{4 u ^{2}} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int \frac{1}{u ^{2}} d u

Apply the Power Rule

: - \frac{1}{u}

= \frac{1}{4} (- \frac{1}{u})

Substitute back

u = 4 sin (t) + 12

= \frac{1}{4} (- \frac{1}{4 sin ( t ) + 12})

Simplify

\frac{1}{4} (- \frac{1}{4 sin ( t ) + 12}) : - \frac{1}{4 ( 4 sin ( t ) + 12 )}

= - \frac{1}{4 ( 4 sin ( t ) + 12 )}

Add a constant to the solution

= - \frac{1}{4 ( 4 sin ( t ) + 12 )} + C

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What is the integral of (cos(t))/((4sin(t)+12)^2) ?
The integral of (cos(t))/((4sin(t)+12)^2) is -1/(4(4sin(t)+12))+C