What is the integral of (cos(x))/(3sin^2(x)+4sin(x)) ?

The integral of (cos(x))/(3sin^2(x)+4sin(x)) is -1/4 (ln| 3/2 sin(x)+2|-ln(3/2 |sin(x)|))+C

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integral of (cos(x))/(3sin^2(x)+4sin(x))

\int \frac{cos ( x )}{3 sin ^{2} ( x ) + 4 sin ( x )} d x

- \frac{1}{4} (ln \frac{3}{2} sin (x) + 2 - ln (\frac{3}{2} ∣ sin (x) ∣)) + C

Solution steps

\int \frac{cos ( x )}{3 sin ^{2} ( x ) + 4 sin ( x )} d x

Apply u-substitution

: \int \frac{1}{3 u ^{2} + 4 u} d u

= \int \frac{1}{3 u ^{2} + 4 u} d u

Complete the square

3 u^{2} + 4 u : 3 (u + \frac{2}{3})^{2} - \frac{4}{3}

= \int \frac{1}{3 ( u + \frac{2}{3} ) ^{2} - \frac{4}{3}} d u

Apply u-substitution

: \int \frac{3}{9 v ^{2} - 4} d v

= \int \frac{3}{9 v ^{2} - 4} d v

Take the constant out:

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

= 3 \cdot \int \frac{1}{9 v ^{2} - 4} d v

Apply Integral Substitution

= 3 \cdot \int \frac{1}{6 ( w ^{2} - 1 )} d w

Take the constant out:

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

= 3 \cdot \frac{1}{6} \cdot \int \frac{1}{w ^{2} - 1} d w

Factor

w^{2} - 1 : - (- w^{2} + 1)

= 3 \cdot \frac{1}{6} \cdot \int \frac{1}{- ( - w ^{2} + 1 )} d w

Take the constant out:

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

= 3 \cdot \frac{1}{6} (- \int \frac{1}{- w ^{2} + 1} d w)

Use the common integral:

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

= 3 \cdot \frac{1}{6} (- (\frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}))

Substitute back

= 3 \cdot \frac{1}{6} (- (\frac{ln \frac{3}{2} ( sin ( x ) + \frac{2}{3} ) + 1}{2} - \frac{ln \frac{3}{2} ( sin ( x ) + \frac{2}{3} ) - 1}{2}))

Simplify

3 \cdot \frac{1}{6} (- (\frac{ln \frac{3}{2} ( sin ( x ) + \frac{2}{3} ) + 1}{2} - \frac{ln \frac{3}{2} ( sin ( x ) + \frac{2}{3} ) - 1}{2})) : - \frac{1}{4} (ln \frac{3}{2} sin (x) + 2 - ln (\frac{3}{2} ∣ sin (x) ∣))

= - \frac{1}{4} (ln \frac{3}{2} sin (x) + 2 - ln (\frac{3}{2} ∣ sin (x) ∣))

Add a constant to the solution

= - \frac{1}{4} (ln \frac{3}{2} sin (x) + 2 - ln (\frac{3}{2} ∣ sin (x) ∣)) + C

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What is the integral of (cos(x))/(3sin^2(x)+4sin(x)) ?
The integral of (cos(x))/(3sin^2(x)+4sin(x)) is -1/4 (ln| 3/2 sin(x)+2|-ln(3/2 |sin(x)|))+C