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

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

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

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

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

Solution steps

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

Simplify

\frac{1}{( x ^{2} + ( 2 ) ^{2} ) ^{1}} : \frac{1}{x ^{2} + 2}

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

Apply Integral Substitution

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

Take the constant out:

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

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

Use the common integral:

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

= 4 \cdot \frac{1}{2} arctan (u)

Substitute back

u = \frac{x}{2}

= 4 \cdot \frac{1}{2} arctan (\frac{x}{2})

Simplify

4 \cdot \frac{1}{2} arctan (\frac{x}{2}) : 22 arctan (\frac{x}{2})

= 22 arctan (\frac{x}{2})

Add a constant to the solution

= 22 arctan (\frac{x}{2}) + C

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