What is the integral of 1/(5+4sin(2x)) ?

The integral of 1/(5+4sin(2x)) is 1/3 arctan(1/3 (5tan(x)+4))+C

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integral of 1/(5+4sin(2x))

\int \frac{1}{5 + 4 sin ( 2 x )} d x

\frac{1}{3} arctan (\frac{1}{3} (5 tan (x) + 4)) + C

Solution steps

\int \frac{1}{5 + 4 sin ( 2 x )} d x

Apply u-substitution

: \int \frac{1}{2 ( 5 + 4 sin ( u ) )} d u

= \int \frac{1}{2 ( 5 + 4 sin ( u ) )} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{1}{5 + 4 sin ( u )} d u

Apply u-substitution

: \int \frac{2}{5 ( 1 + v ^{2} ) + 8 v} d v

= \frac{1}{2} \cdot \int \frac{2}{5 ( 1 + v ^{2} ) + 8 v} d v

Take the constant out:

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

= \frac{1}{2} \cdot 2 \cdot \int \frac{1}{5 ( 1 + v ^{2} ) + 8 v} d v

Expand

5 (1 + v^{2}) : 5 + 5 v^{2}

= \frac{1}{2} \cdot 2 \cdot \int \frac{1}{5 + 5 v ^{2} + 8 v} d v

Complete the square

5 + 5 v^{2} + 8 v : 5 (v + \frac{4}{5})^{2} + \frac{9}{5}

= \frac{1}{2} \cdot 2 \cdot \int \frac{1}{5 ( v + \frac{4}{5} ) ^{2} + \frac{9}{5}} d v

Apply u-substitution

: \int \frac{5}{25 w ^{2} + 9} d w

= \frac{1}{2} \cdot 2 \cdot \int \frac{5}{25 w ^{2} + 9} d w

Take the constant out:

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

= \frac{1}{2} \cdot 2 \cdot 5 \cdot \int \frac{1}{25 w ^{2} + 9} d w

Apply Integral Substitution

= \frac{1}{2} \cdot 2 \cdot 5 \cdot \int \frac{1}{15 ( t ^{2} + 1 )} d t

Take the constant out:

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

= \frac{1}{2} \cdot 2 \cdot 5 \cdot \frac{1}{15} \cdot \int \frac{1}{t ^{2} + 1} d t

Use the common integral:

\int \frac{1}{t ^{2} + 1} d t = arctan (t)

= \frac{1}{2} \cdot 2 \cdot 5 \cdot \frac{1}{15} arctan (t)

Substitute back

= \frac{1}{2} \cdot 2 \cdot 5 \cdot \frac{1}{15} arctan (\frac{5}{3} (tan (\frac{2 x}{2}) + \frac{4}{5}))

Simplify

\frac{1}{2} \cdot 2 \cdot 5 \cdot \frac{1}{15} arctan (\frac{5}{3} (tan (\frac{2 x}{2}) + \frac{4}{5})) : \frac{1}{3} arctan (\frac{1}{3} (5 tan (x) + 4))

= \frac{1}{3} arctan (\frac{1}{3} (5 tan (x) + 4))

Add a constant to the solution

= \frac{1}{3} arctan (\frac{1}{3} (5 tan (x) + 4)) + C

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What is the integral of 1/(5+4sin(2x)) ?
The integral of 1/(5+4sin(2x)) is 1/3 arctan(1/3 (5tan(x)+4))+C