What is the integral of (4sin(x))/(3+cos(x)) ?

The integral of (4sin(x))/(3+cos(x)) is -4ln|3+cos(x)|+C

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integral of (4sin(x))/(3+cos(x))

\int \frac{4 sin ( x )}{3 + cos ( x )} d x

- 4 ln ∣ 3 + cos (x) ∣ + C

Solution steps

\int \frac{4 sin ( x )}{3 + cos ( x )} d x

Take the constant out:

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

= 4 \cdot \int \frac{sin ( x )}{3 + cos ( x )} d x

Apply u-substitution

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

= 4 \cdot \int - \frac{1}{u} d u

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 4 (- ln ∣ u ∣)

Substitute back

u = 3 + cos (x)

= 4 (- ln ∣ 3 + cos (x) ∣)

Simplify

= - 4 ln ∣ 3 + cos (x) ∣

Add a constant to the solution

= - 4 ln ∣ 3 + cos (x) ∣ + C

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What is the integral of (4sin(x))/(3+cos(x)) ?
The integral of (4sin(x))/(3+cos(x)) is -4ln|3+cos(x)|+C