What is the integral from 2 to 5 of x/(x^2+4x+20) ?

The integral from 2 to 5 of x/(x^2+4x+20) is 1/2 (ln(65/16)-ln(2)-arctan(7/4)+pi/4)

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integral from 2 to 5 of x/(x^2+4x+20)

\int_{2}^{5} \frac{x}{x ^{2} + 4 x + 20} d x

\frac{1}{2} (ln (\frac{65}{16}) - ln (2) - arctan (\frac{7}{4}) + \frac{π}{4})

Decimal Notation

0.22119 \dots

Solution steps

\int_{2}^{5} \frac{x}{x ^{2} + 4 x + 20} d x

Complete the square

x^{2} + 4 x + 20 : (x + 2)^{2} + 16

= \int_{2}^{5} \frac{x}{( x + 2 ) ^{2} + 16} d x

Apply u-substitution

: \int_{4}^{7} \frac{u - 2}{u ^{2} + 16} d u

= \int_{4}^{7} \frac{u - 2}{u ^{2} + 16} d u

Apply Integral Substitution

= \int_{1}^{\frac{7}{4}} \frac{2 v - 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_{1}^{\frac{7}{4}} \frac{2 v - 1}{v ^{2} + 1} d v

Expand

\frac{2 v - 1}{v ^{2} + 1} : \frac{2 v}{v ^{2} + 1} - \frac{1}{v ^{2} + 1}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int_{1}^{\frac{7}{4}} \frac{2 v}{v ^{2} + 1} d v - \int_{1}^{\frac{7}{4}} \frac{1}{v ^{2} + 1} d v)

\int_{1}^{\frac{7}{4}} \frac{2 v}{v ^{2} + 1} d v = ln (\frac{65}{16}) - ln (2)

\int_{1}^{\frac{7}{4}} \frac{1}{v ^{2} + 1} d v = arctan (\frac{7}{4}) - \frac{π}{4}

= \frac{1}{2} (ln (\frac{65}{16}) - ln (2) - (arctan (\frac{7}{4}) - \frac{π}{4}))

Simplify

= \frac{1}{2} (ln (\frac{65}{16}) - ln (2) - arctan (\frac{7}{4}) + \frac{π}{4})

(x^{2} - 16 x + 64) (x^{2} + 64 x + 4)

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

\frac{d y}{d x} = (\frac{x - y}{x})

(cos (5 x))^{^{'}}

f (x) = \frac{6 x}{x + 5}

What is the integral from 2 to 5 of x/(x^2+4x+20) ?
The integral from 2 to 5 of x/(x^2+4x+20) is 1/2 (ln(65/16)-ln(2)-arctan(7/4)+pi/4)