What is the integral of 1/(cos^2(x)(3tan(x)+1)) ?

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

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integral of 1/(cos^2(x)(3tan(x)+1))

\int \frac{1}{cos ^{2} ( x ) ( 3 tan ( x ) + 1 )} d x

\frac{1}{3} ln ∣ 3 tan (x) + 1 ∣ + C

Solution steps

\int \frac{1}{cos ^{2} ( x ) ( 3 tan ( x ) + 1 )} d x

Apply u-substitution

: \int \frac{1}{3 u} d u

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

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}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{3} ln ∣ u ∣

Substitute back

u = 3 tan (x) + 1

= \frac{1}{3} ln ∣ 3 tan (x) + 1 ∣

Add a constant to the solution

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

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