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

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

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

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

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

Solution steps

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

Apply u-substitution

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

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

Apply Integral Substitution

= \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

= \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)

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

Substitute back

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

Add a constant to the solution

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

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

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

\int \frac{x ^{2}}{( a x ^{3} + b ) ^{2}} d x

\int \frac{tan ( x )}{( cos ( x ) ) ^{2}} d x

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