What is the integral of (3+sin(4x))/(sin(4x)) ?

The integral of (3+sin(4x))/(sin(4x)) is 3/4 ln|tan(2x)|+x+C

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integral of (3+sin(4x))/(sin(4x))

\int \frac{3 + sin ( 4 x )}{sin ( 4 x )} d x

\frac{3}{4} ln ∣ tan (2 x) ∣ + x + C

Solution steps

\int \frac{3 + sin ( 4 x )}{sin ( 4 x )} d x

Expand

\frac{3 + sin ( 4 x )}{sin ( 4 x )} : \frac{3}{sin ( 4 x )} + 1

= \int \frac{3}{sin ( 4 x )} + 1 d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{3}{sin ( 4 x )} d x + \int 1 d x

\int \frac{3}{sin ( 4 x )} d x = \frac{3}{4} ln ∣ tan (2 x) ∣

\int 1 d x = x

= \frac{3}{4} ln ∣ tan (2 x) ∣ + x

Add a constant to the solution

= \frac{3}{4} ln ∣ tan (2 x) ∣ + x + C

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What is the integral of (3+sin(4x))/(sin(4x)) ?
The integral of (3+sin(4x))/(sin(4x)) is 3/4 ln|tan(2x)|+x+C