What is the integral from 1 to 5 of 2/(x^2+2x+1) ?

The integral from 1 to 5 of 2/(x^2+2x+1) is 2/3

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integral from 1 to 5 of 2/(x^2+2x+1)

\int_{1}^{5} \frac{2}{x ^{2} + 2 x + 1} d x

\frac{2}{3}

Decimal Notation

0.66666 \dots

Solution steps

\int_{1}^{5} \frac{2}{x ^{2} + 2 x + 1} d x

Take the constant out:

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

= 2 \cdot \int_{1}^{5} \frac{1}{x ^{2} + 2 x + 1} d x

Complete the square

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

= 2 \cdot \int_{1}^{5} \frac{1}{( x + 1 ) ^{2}} d x

Apply u-substitution

: \int_{2}^{6} \frac{1}{u ^{2}} d u

= 2 \cdot \int_{2}^{6} \frac{1}{u ^{2}} d u

Apply the Power Rule

: [- \frac{1}{u}]_{2}^{6}

= 2 [- \frac{1}{u}]_{2}^{6}

Compute the boundaries:

\frac{1}{3}

= 2 \cdot \frac{1}{3}

Simplify

= \frac{2}{3}

\int e^{- s t} \cdot (t^{2} - 3 t + 1) d t

ln (e^{x} + x e^{x})

\frac{d}{d x} ((3 x + x) (2 - x^{2}))

\int_{1}^{6} 8 x d x

\frac{5 x}{1 + x ^{2}}

What is the integral from 1 to 5 of 2/(x^2+2x+1) ?
The integral from 1 to 5 of 2/(x^2+2x+1) is 2/3