What is the integral of 1/(3sin(x)+4cos(x)+4) ?

The integral of 1/(3sin(x)+4cos(x)+4) is 1/3 ln|3tan(x/2)+4|+C

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integral of 1/(3sin(x)+4cos(x)+4)

\int \frac{1}{3 sin ( x ) + 4 cos ( x ) + 4} d x

\frac{1}{3} ln 3 tan (\frac{x}{2}) + 4 + C

Solution steps

\int \frac{1}{3 sin ( x ) + 4 cos ( x ) + 4} d x

Apply u-substitution

= \int \frac{1}{3 u + 4} d u

Apply u-substitution

= \int \frac{1}{3 v} d v

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{3} ln ∣ v ∣

Substitute back

= \frac{1}{3} ln 3 tan (\frac{x}{2}) + 4

Add a constant to the solution

= \frac{1}{3} ln 3 tan (\frac{x}{2}) + 4 + C

What is the integral of 1/(3sin(x)+4cos(x)+4) ?
The integral of 1/(3sin(x)+4cos(x)+4) is 1/3 ln|3tan(x/2)+4|+C