What is the integral from 5 to 6 of x/(x^2+4x+20) ?

The integral from 5 to 6 of x/(x^2+4x+20) is 1/2 (ln(16/13)-arctan(2)+arctan(7/4))

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integral from 5 to 6 of x/(x^2+4x+20)

\int_{5}^{6} \frac{x}{x ^{2} + 4 x + 20} d x

\frac{1}{2} (ln (\frac{16}{13}) - arctan (2) + arctan (\frac{7}{4}))

Decimal Notation

0.07607 \dots

Solution steps

\int_{5}^{6} \frac{x}{x ^{2} + 4 x + 20} d x

Complete the square

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

= \int_{5}^{6} \frac{x}{( x + 2 ) ^{2} + 16} d x

Apply u-substitution

: \int_{7}^{8} \frac{u - 2}{u ^{2} + 16} d u

= \int_{7}^{8} \frac{u - 2}{u ^{2} + 16} d u

Apply Integral Substitution

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

\int_{\frac{7}{4}}^{2} \frac{2 v}{v ^{2} + 1} d v = ln (\frac{16}{13})

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

= \frac{1}{2} (ln (\frac{16}{13}) - (arctan (2) - arctan (\frac{7}{4})))

Simplify

= \frac{1}{2} (ln (\frac{16}{13}) - arctan (2) + arctan (\frac{7}{4}))

y^{^{''}} + 4 y^{^{'}} + 5 y = e^{x}

\frac{d}{d x} (- 8 e^{- 8 x})

\frac{\partial}{\partial r} (r^{2} + s^{2})

x \to 3 - lim (\frac{5 x}{x ^{2} - 9})

\int_{0}^{5} \frac{1}{25 - x ^{2}} d x

What is the integral from 5 to 6 of x/(x^2+4x+20) ?
The integral from 5 to 6 of x/(x^2+4x+20) is 1/2 (ln(16/13)-arctan(2)+arctan(7/4))