What is the integral of 1/(cos(x)+3) ?

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

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integral of 1/(cos(x)+3)

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

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

Solution steps

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

Apply u-substitution

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

Apply Integral Substitution

= \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

= \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)

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

Substitute back

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

Add a constant to the solution

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

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