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

The integral of 2/(3+cos(x)) is sqrt(2)arctan(1/(sqrt(2))tan(x/2))+C

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integral of 2/(3+cos(x))

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

2 arctan (\frac{1}{2} tan (\frac{x}{2})) + C

Solution steps

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

Take the constant out:

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

= 2 \cdot \int \frac{1}{3 + cos ( x )} d x

Apply u-substitution

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

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

Apply Integral Substitution

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

Take the constant out:

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

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

Use the common integral:

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

= 2 \cdot \frac{1}{2} arctan (v)

Substitute back

= 2 \cdot \frac{1}{2} arctan (\frac{tan ( \frac{x}{2} )}{2})

Simplify

2 \cdot \frac{1}{2} arctan (\frac{tan ( \frac{x}{2} )}{2}) : 2 arctan (\frac{1}{2} tan (\frac{x}{2}))

= 2 arctan (\frac{1}{2} tan (\frac{x}{2}))

Add a constant to the solution

= 2 arctan (\frac{1}{2} tan (\frac{x}{2})) + C

\int \frac{1}{x ^{2}} \cdot cos (\frac{1}{x}) d x

\int (x \cdot 2^{x}) d x

\int (\frac{1}{x ^{2} + a ^{2}}) d x

\int \frac{4}{x ( ln ( x ) + 1 ) ^{2}} d x

\int - x^{3} + 2 x^{2} + x - 2 d x

What is the integral of 2/(3+cos(x)) ?
The integral of 2/(3+cos(x)) is sqrt(2)arctan(1/(sqrt(2))tan(x/2))+C