What is the integral of-(4cos(x))/(3-10sin(x)) ?

The integral of-(4cos(x))/(3-10sin(x)) is 2/5 ln|3-10sin(x)|+C

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integral of-(4cos(x))/(3-10sin(x))

\int - \frac{4 cos ( x )}{3 - 10 sin ( x )} d x

\frac{2}{5} ln ∣ 3 - 10 sin (x) ∣ + C

Solution steps

\int - \frac{4 cos ( x )}{3 - 10 sin ( 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{cos ( x )}{3 - 10 sin ( x )} d x

Apply u-substitution

= - 4 \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

= - 4 (- \frac{1}{10} \cdot \int \frac{1}{u} d u)

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - 4 (- \frac{1}{10} ln ∣ u ∣)

Substitute back

u = 3 - 10 sin (x)

= - 4 (- \frac{1}{10} ln ∣ 3 - 10 sin (x) ∣)

Simplify

- 4 (- \frac{1}{10} ln ∣ 3 - 10 sin (x) ∣) : \frac{2}{5} ln ∣ 3 - 10 sin (x) ∣

= \frac{2}{5} ln ∣ 3 - 10 sin (x) ∣

Add a constant to the solution

= \frac{2}{5} ln ∣ 3 - 10 sin (x) ∣ + C

What is the integral of-(4cos(x))/(3-10sin(x)) ?
The integral of-(4cos(x))/(3-10sin(x)) is 2/5 ln|3-10sin(x)|+C