What is the integral from-2 to 0 of 1/(x^2+2x+2) ?

The integral from-2 to 0 of 1/(x^2+2x+2) is pi/2

integral from-2 to 0 of 1/(x^2+2x+2)

\int_{- 2}^{0} \frac{1}{x ^{2} + 2 x + 2} d x

\frac{π}{2}

Decimal

1.57079 \dots

Solution steps

\int_{- 2}^{0} \frac{1}{x ^{2} + 2 x + 2} d x

Complete the square

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

= \int_{- 2}^{0} \frac{1}{( x + 1 ) ^{2} + 1} d x

Apply u-substitution

= \int_{- 1}^{1} \frac{1}{u ^{2} + 1} d u

Use the common integral:

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

= [arctan (u)]_{- 1}^{1}

Compute the boundaries:

\frac{π}{2}

= \frac{π}{2}

\int_{- 1}^{0} \frac{e ^{\frac{1}{x}}}{x ^{3}} d x

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

\int_{0}^{\frac{π}{4}} cos (x) - sin (x) d x

\int_{0}^{\frac{π}{2}} cos (x) sin (x) d x

\int_{0}^{π} (1 - sin^{3} (x)) d x

What is the integral from-2 to 0 of 1/(x^2+2x+2) ?
The integral from-2 to 0 of 1/(x^2+2x+2) is pi/2