What is the integral of (sin(t))/((4+cos(t))^5) ?

The integral of (sin(t))/((4+cos(t))^5) is 1/(4(4+cos(t))^4)+C

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integral of (sin(t))/((4+cos(t))^5)

\int \frac{sin ( t )}{( 4 + cos ( t ) ) ^{5}} d t

\frac{1}{4 ( 4 + cos ( t ) ) ^{4}} + C

Solution steps

\int \frac{sin ( t )}{( 4 + cos ( t ) ) ^{5}} d t

Apply u-substitution

: \int - \frac{1}{u ^{5}} d u

= \int - \frac{1}{u ^{5}} d u

Take the constant out:

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

= - \int \frac{1}{u ^{5}} d u

Apply the Power Rule

: - \frac{1}{4 u ^{4}}

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

Substitute back

u = 4 + cos (t)

= - (- \frac{1}{4 ( 4 + cos ( t ) ) ^{4}})

Simplify

= \frac{1}{4 ( 4 + cos ( t ) ) ^{4}}

Add a constant to the solution

= \frac{1}{4 ( 4 + cos ( t ) ) ^{4}} + C

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What is the integral of (sin(t))/((4+cos(t))^5) ?
The integral of (sin(t))/((4+cos(t))^5) is 1/(4(4+cos(t))^4)+C