integral of 1/(5sin^2(x)-3cos^2(x)-2)

Solution

\int \frac{1}{5 sin ^{2} ( x ) - 3 cos ^{2} ( x ) - 2} d x

- \frac{1}{2 15} (ln \frac{5}{3} cot (x) + 1 - ln \frac{5}{3} cot (x) - 1) + C

Solution steps

\int \frac{1}{5 sin ^{2} ( x ) - 3 cos ^{2} ( x ) - 2} d x

Rewrite using trig identities

= \int \frac{1}{- 8 cos ^{2} ( x ) + 3} d x

Apply Trigonometric Substitution

: \int - \frac{1}{- 5 u ^{2} + 3} d u

= \int - \frac{1}{- 5 u ^{2} + 3} d u

Take the constant out:

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

= - \int \frac{1}{- 5 u ^{2} + 3} d u

Apply Integral Substitution

= - \int \frac{1}{15 ( - v ^{2} + 1 )} d v

Take the constant out:

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

= - \frac{1}{15} \cdot \int \frac{1}{- v ^{2} + 1} d v

Use the common integral:

\int \frac{1}{- v ^{2} + 1} d v = \frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}

= - \frac{1}{15} (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2})

Substitute back

= - \frac{1}{15} \frac{ln \frac{5}{3} cot ( x ) + 1}{2} - \frac{ln \frac{5}{3} cot ( x ) - 1}{2}

Simplify

- \frac{1}{15} \frac{ln \frac{5}{3} cot ( x ) + 1}{2} - \frac{ln \frac{5}{3} cot ( x ) - 1}{2} : - \frac{1}{2 15} (ln \frac{5}{3} cot (x) + 1 - ln \frac{5}{3} cot (x) - 1)

= - \frac{1}{2 15} (ln \frac{5}{3} cot (x) + 1 - ln \frac{5}{3} cot (x) - 1)

Add a constant to the solution

= - \frac{1}{2 15} (ln \frac{5}{3} cot (x) + 1 - ln \frac{5}{3} cot (x) - 1) + C

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