What is the integral of 4/(1+(x+2)^2) ?

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

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integral of 4/(1+(x+2)^2)

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

4 arctan (x + 2) + C

Solution steps

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

Take the constant out:

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

= 4 \cdot \int \frac{1}{1 + ( x + 2 ) ^{2}} d x

Apply u-substitution

: \int \frac{1}{1 + u ^{2}} d u

= 4 \cdot \int \frac{1}{1 + u ^{2}} d u

Use the common integral:

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

= 4 arctan (u)

Substitute back

u = x + 2

= 4 arctan (x + 2)

Add a constant to the solution

= 4 arctan (x + 2) + C

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