What is the integral of 1/(sin(x)(sin(x)+cos(x))) ?

The integral of 1/(sin(x)(sin(x)+cos(x))) is -ln|1+cot(x)|+C

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integral of 1/(sin(x)(sin(x)+cos(x)))

\int \frac{1}{sin ( x ) ( sin ( x ) + cos ( x ) )} d x

- ln ∣ 1 + cot (x) ∣ + C

Solution steps

\int \frac{1}{sin ( x ) ( sin ( x ) + cos ( x ) )} d x

Expand

sin (x) (sin (x) + cos (x)) : sin^{2} (x) + sin (x) cos (x)

= \int \frac{1}{sin ^{2} ( x ) + sin ( x ) cos ( x )} d x

Apply Trigonometric Substitution

: \int - \frac{1}{1 + u} d u

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{1}{v} d v

= - \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= - ln ∣ v ∣

Substitute back

= - ln ∣ 1 + cot (x) ∣

Add a constant to the solution

= - ln ∣ 1 + cot (x) ∣ + C

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