What is the integral of (7cos(x))/(2sin(x)+1) ?

The integral of (7cos(x))/(2sin(x)+1) is 7/2 ln|2sin(x)+1|+C

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integral of (7cos(x))/(2sin(x)+1)

\int \frac{7 cos ( x )}{2 sin ( x ) + 1} d x

\frac{7}{2} ln ∣ 2 sin (x) + 1 ∣ + C

Solution steps

\int \frac{7 cos ( x )}{2 sin ( x ) + 1} d x

Take the constant out:

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

= 7 \cdot \int \frac{cos ( x )}{2 sin ( x ) + 1} d x

Apply u-substitution

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

= 7 \cdot \int \frac{1}{2 u} d u

Take the constant out:

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

= 7 \cdot \frac{1}{2} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 7 \cdot \frac{1}{2} ln ∣ u ∣

Substitute back

u = 2 sin (x) + 1

= 7 \cdot \frac{1}{2} ln ∣ 2 sin (x) + 1 ∣

Simplify

7 \cdot \frac{1}{2} ln ∣ 2 sin (x) + 1 ∣ : \frac{7}{2} ln ∣ 2 sin (x) + 1 ∣

= \frac{7}{2} ln ∣ 2 sin (x) + 1 ∣

Add a constant to the solution

= \frac{7}{2} ln ∣ 2 sin (x) + 1 ∣ + C

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What is the integral of (7cos(x))/(2sin(x)+1) ?
The integral of (7cos(x))/(2sin(x)+1) is 7/2 ln|2sin(x)+1|+C