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

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

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

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

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

Solution steps

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

Apply u-substitution

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

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

Apply Integral Substitution

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

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{1}{v ^{2} + 1} d v

Use the common integral:

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

= 2 \cdot \frac{1}{2} arctan (v)

Substitute back

= 2 \cdot \frac{1}{2} arctan (\frac{x - 2}{2})

Simplify

2 \cdot \frac{1}{2} arctan (\frac{x - 2}{2}) : arctan (\frac{1}{2} (x - 2))

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

Add a constant to the solution

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

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