What is the integral of 1/(1+(2x+5)^2) ?

The integral of 1/(1+(2x+5)^2) is 1/2 arctan(2x+5)+C

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integral of 1/(1+(2x+5)^2)

\int \frac{1}{1 + ( 2 x + 5 ) ^{2}} d x

\frac{1}{2} arctan (2 x + 5) + C

Solution steps

\int \frac{1}{1 + ( 2 x + 5 ) ^{2}} d x

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

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

= \frac{1}{2} arctan (u)

Substitute back

u = 2 x + 5

= \frac{1}{2} arctan (2 x + 5)

Add a constant to the solution

= \frac{1}{2} arctan (2 x + 5) + C

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What is the integral of 1/(1+(2x+5)^2) ?
The integral of 1/(1+(2x+5)^2) is 1/2 arctan(2x+5)+C