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

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

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

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

ln ∣ sin (x) + 3 x ∣ + C

Solution steps

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

Apply u-substitution

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= ln ∣ u ∣

Substitute back

u = sin (x) + 3 x

= ln ∣ sin (x) + 3 x ∣

Add a constant to the solution

= ln ∣ sin (x) + 3 x ∣ + C

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