What is the integral of 2arctan(x+2) ?

The integral of 2arctan(x+2) is 2xarctan(x+2)+4arctan(x+2)-ln|x^2+4x+5|+C

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integral of 2arctan(x+2)

\int 2 arctan (x + 2) d x

2 x arctan (x + 2) + 4 arctan (x + 2) - ln x^{2} + 4 x + 5 + C

Solution steps

\int 2 arctan (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 arctan (x + 2) d x

Apply u-substitution

: \int arctan (u) d u

= 2 \cdot \int arctan (u) d u

Apply Integration By Parts

: u arctan (u) - \int \frac{u}{u ^{2} + 1} d u

= 2 (u arctan (u) - \int \frac{u}{u ^{2} + 1} d u)

\int \frac{u}{u ^{2} + 1} d u = \frac{1}{2} ln u^{2} + 1

= 2 (u arctan (u) - \frac{1}{2} ln u^{2} + 1)

Substitute back

u = x + 2

= 2 ((x + 2) arctan (x + 2) - \frac{1}{2} ln (x + 2)^{2} + 1)

Simplify

2 ((x + 2) arctan (x + 2) - \frac{1}{2} ln (x + 2)^{2} + 1) : 2 x arctan (x + 2) + 4 arctan (x + 2) - ln x^{2} + 4 x + 5

= 2 x arctan (x + 2) + 4 arctan (x + 2) - ln x^{2} + 4 x + 5

Add a constant to the solution

= 2 x arctan (x + 2) + 4 arctan (x + 2) - ln x^{2} + 4 x + 5 + C

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What is the integral of 2arctan(x+2) ?
The integral of 2arctan(x+2) is 2xarctan(x+2)+4arctan(x+2)-ln|x^2+4x+5|+C