What is the integral of (cos(5x))/(3+sin(5x)) ?

The integral of (cos(5x))/(3+sin(5x)) is 1/5 ln|3+sin(5x)|+C

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integral of (cos(5x))/(3+sin(5x))

\int \frac{cos ( 5 x )}{3 + sin ( 5 x )} d x

\frac{1}{5} ln ∣ 3 + sin (5 x) ∣ + C

Solution steps

\int \frac{cos ( 5 x )}{3 + sin ( 5 x )} d x

Apply u-substitution

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

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

Take the constant out:

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

= \frac{1}{5} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{5} ln ∣ u ∣

Substitute back

u = 3 + sin (5 x)

= \frac{1}{5} ln ∣ 3 + sin (5 x) ∣

Add a constant to the solution

= \frac{1}{5} ln ∣ 3 + sin (5 x) ∣ + C

y^{^{''}} + 25 y = 5 (csc (5 t))^{2}

t y^{^{'}} + (t + 1) y = t, y (ln (6)) = 1

\frac{\partial}{\partial x} (3 x^{2} + 7 y^{2})

(2.3) (4.7)

\int - ln ∣ 1 - x ∣ d x

What is the integral of (cos(5x))/(3+sin(5x)) ?
The integral of (cos(5x))/(3+sin(5x)) is 1/5 ln|3+sin(5x)|+C