What is the integral of (e^{2θ})/(1+3e^{2θ)} ?

The integral of (e^{2θ})/(1+3e^{2θ)} is 1/6 ln|1+3e^{2θ}|+C

Popular >

integral of (e^{2θ})/(1+3e^{2θ)}

\int \frac{e ^{2 θ}}{1 + 3 e ^{2 θ}} d θ

\frac{1}{6} ln 1 + 3 e^{2 θ} + C

Solution steps

\int \frac{e ^{2 θ}}{1 + 3 e ^{2 θ}} d θ

Apply u-substitution

: \int \frac{1}{6 u} d u

= \int \frac{1}{6 u} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{6} ln ∣ u ∣

Substitute back

u = 1 + 3 e^{2 θ}

= \frac{1}{6} ln 1 + 3 e^{2 θ}

Add a constant to the solution

= \frac{1}{6} ln 1 + 3 e^{2 θ} + C

\int \frac{x ^{2} + x + 1}{x ^{4} + 2 x ^{2} + 1} d x

\int \frac{x}{x ^{2} x ^{2} + 1} d x

\int ((8 - x) (8 + x)) d x

\int x^{3} ln (x) + cos (3 x + 2) d x

\int e^{\frac{2}{3} x} + 4 x d x

What is the integral of (e^{2θ})/(1+3e^{2θ)} ?
The integral of (e^{2θ})/(1+3e^{2θ)} is 1/6 ln|1+3e^{2θ}|+C