What is the integral of (3cos(x))/(-2+10sin(x)) ?

The integral of (3cos(x))/(-2+10sin(x)) is 3/10 ln|-2+10sin(x)|+C

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integral of (3cos(x))/(-2+10sin(x))

\int \frac{3 cos ( x )}{- 2 + 10 sin ( x )} d x

\frac{3}{10} ln ∣ - 2 + 10 sin (x) ∣ + C

Solution steps

\int \frac{3 cos ( x )}{- 2 + 10 sin ( x )} d x

Take the constant out:

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

= 3 \cdot \int \frac{cos ( x )}{- 2 + 10 sin ( x )} d x

Apply u-substitution

: \int \frac{1}{10 u} d u

= 3 \cdot \int \frac{1}{10 u} d u

Take the constant out:

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

= 3 \cdot \frac{1}{10} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 3 \cdot \frac{1}{10} ln ∣ u ∣

Substitute back

u = - 2 + 10 sin (x)

= 3 \cdot \frac{1}{10} ln ∣ - 2 + 10 sin (x) ∣

Simplify

3 \cdot \frac{1}{10} ln ∣ - 2 + 10 sin (x) ∣ : \frac{3}{10} ln ∣ - 2 + 10 sin (x) ∣

= \frac{3}{10} ln ∣ - 2 + 10 sin (x) ∣

Add a constant to the solution

= \frac{3}{10} ln ∣ - 2 + 10 sin (x) ∣ + C

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What is the integral of (3cos(x))/(-2+10sin(x)) ?
The integral of (3cos(x))/(-2+10sin(x)) is 3/10 ln|-2+10sin(x)|+C