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

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

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

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

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

Solution steps

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

Apply u-substitution

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

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

Use the common integral:

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

= 2 arctan (u)

Substitute back

u = \frac{x}{2}

= 2 arctan (\frac{x}{2})

Add a constant to the solution

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

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