What is the integral of 1/(1+cos(10x)) ?

The integral of 1/(1+cos(10x)) is 1/10 tan(5x)+C

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integral of 1/(1+cos(10x))

\int \frac{1}{1 + cos ( 10 x )} d x

\frac{1}{10} tan (5 x) + C

Solution steps

\int \frac{1}{1 + cos ( 10 x )} d x

Apply u-substitution

= \int \frac{1}{10 ( 1 + cos ( u ) )} d u

Take the constant out:

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

= \frac{1}{10} \cdot \int \frac{1}{1 + cos ( u )} d u

Apply u-substitution

= \frac{1}{10} \cdot \int 1 d v

Integral of a constant:

\int a d x = a x

= \frac{1}{10} \cdot 1 \cdot v

Substitute back

= \frac{1}{10} \cdot 1 \cdot tan (\frac{10 x}{2})

Simplify

\frac{1}{10} \cdot 1 \cdot tan (\frac{10 x}{2}) : \frac{1}{10} tan (5 x)

= \frac{1}{10} tan (5 x)

Add a constant to the solution

= \frac{1}{10} tan (5 x) + C

What is the integral of 1/(1+cos(10x)) ?
The integral of 1/(1+cos(10x)) is 1/10 tan(5x)+C