What is the integral of (2e^{arctan(2x)})/(1+4x^2) ?

The integral of (2e^{arctan(2x)})/(1+4x^2) is e^{arctan(2x)}+C

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integral of (2e^{arctan(2x)})/(1+4x^2)

\int \frac{2 e ^{a r c t a n (2 x)}}{1 + 4 x ^{2}} d x

e^{a r c t a n (2 x)} + C

Solution steps

\int \frac{2 e ^{a r c t a n (2 x)}}{1 + 4 x ^{2}} d x

Take the constant out:

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

= 2 \cdot \int \frac{e ^{a r c t a n (2 x)}}{1 + 4 x ^{2}} d x

Apply Integral Substitution

= 2 \cdot \int \frac{e ^{a r c t a n (u)}}{2 ( 1 + u ^{2} )} d u

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot \int \frac{e ^{a r c t a n (u)}}{1 + u ^{2}} d u

Apply u-substitution

: \int e^{v} d v

= 2 \cdot \frac{1}{2} \cdot \int e^{v} d v

Use the common integral:

\int e^{v} d v = e^{v}

= 2 \cdot \frac{1}{2} e^{v}

Substitute back

= 2 \cdot \frac{1}{2} e^{a r c t a n (2 x)}

Simplify

2 \cdot \frac{1}{2} e^{a r c t a n (2 x)} : e^{a r c t a n (2 x)}

= e^{a r c t a n (2 x)}

Add a constant to the solution

= e^{a r c t a n (2 x)} + C

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What is the integral of (2e^{arctan(2x)})/(1+4x^2) ?
The integral of (2e^{arctan(2x)})/(1+4x^2) is e^{arctan(2x)}+C