What is the integral of 1/(sin^2(x)-5sin(x)cos(x)) ?

The integral of 1/(sin^2(x)-5sin(x)cos(x)) is 1/5 ln|1-5cot(x)|+C

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integral of 1/(sin^2(x)-5sin(x)cos(x))

\int \frac{1}{sin ^{2} ( x ) - 5 sin ( x ) cos ( x )} d x

\frac{1}{5} ln ∣ 1 - 5 cot (x) ∣ + C

Solution steps

\int \frac{1}{sin ^{2} ( x ) - 5 sin ( x ) cos ( x )} d x

Apply Trigonometric Substitution

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

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

Apply u-substitution

: \int - \frac{1}{5 v} d v

= - \int - \frac{1}{5 v} d v

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}{v} d v)

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= - (- \frac{1}{5} ln ∣ v ∣)

Substitute back

= - (- \frac{1}{5} ln ∣ 1 - 5 cot (x) ∣)

Simplify

= \frac{1}{5} ln ∣ 1 - 5 cot (x) ∣

Add a constant to the solution

= \frac{1}{5} ln ∣ 1 - 5 cot (x) ∣ + C

\int (e^{1 - 3 x} + 2 cos (\frac{x}{9})) d x

\int x ln (x^{3} + 1) d x

\int (x - 1) (2 x + 1) d x

What is the integral of 1/(sin^2(x)-5sin(x)cos(x)) ?
The integral of 1/(sin^2(x)-5sin(x)cos(x)) is 1/5 ln|1-5cot(x)|+C