What is the integral of 1/(1+cos(x)+7sin(x)) ?

The integral of 1/(1+cos(x)+7sin(x)) is 1/7 ln|7tan(x/2)+1|+C

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integral of 1/(1+cos(x)+7sin(x))

\int \frac{1}{1 + cos ( x ) + 7 sin ( x )} d x

\frac{1}{7} ln 7 tan (\frac{x}{2}) + 1 + C

Solution steps

\int \frac{1}{1 + cos ( x ) + 7 sin ( x )} d x

Apply u-substitution

: \int \frac{1}{7 u + 1} d u

= \int \frac{1}{7 u + 1} d u

Apply u-substitution

: \int \frac{1}{7 v} d v

= \int \frac{1}{7 v} d v

Take the constant out:

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

= \frac{1}{7} \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{7} ln ∣ v ∣

Substitute back

= \frac{1}{7} ln 7 tan (\frac{x}{2}) + 1

Add a constant to the solution

= \frac{1}{7} ln 7 tan (\frac{x}{2}) + 1 + C

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What is the integral of 1/(1+cos(x)+7sin(x)) ?
The integral of 1/(1+cos(x)+7sin(x)) is 1/7 ln|7tan(x/2)+1|+C