What is the integral from 7 to 8 of x/(x^2+6x+13) ?

The integral from 7 to 8 of x/(x^2+6x+13) is (3ln(5)-ln(104)-3(arctan(11/2)-arctan(5)))/2

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integral from 7 to 8 of x/(x^2+6x+13)

\int_{7}^{8} \frac{x}{x ^{2} + 6 x + 13} d x

\frac{3 ln ( 5 ) - ln ( 104 ) - 3 ( arctan ( \frac{11}{2} ) - arctan ( 5 ) )}{2}

Decimal

0.06564 \dots

Solution steps

\int_{7}^{8} \frac{x}{x ^{2} + 6 x + 13} d x

Complete the square

x^{2} + 6 x + 13 : (x + 3)^{2} + 4

= \int_{7}^{8} \frac{x}{( x + 3 ) ^{2} + 4} d x

Apply u-substitution

= \int_{10}^{11} \frac{u - 3}{u ^{2} + 4} d u

Expand

\frac{u - 3}{u ^{2} + 4} : \frac{u}{u ^{2} + 4} - \frac{3}{u ^{2} + 4}

Apply the Sum Rule:

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

= \int_{10}^{11} \frac{u}{u ^{2} + 4} d u - \int_{10}^{11} \frac{3}{u ^{2} + 4} d u

\int_{10}^{11} \frac{u}{u ^{2} + 4} d u = \frac{1}{2} (3 ln (5) - ln (104))

\int_{10}^{11} \frac{3}{u ^{2} + 4} d u = \frac{3}{2} (arctan (\frac{11}{2}) - arctan (5))

= \frac{1}{2} (3 ln (5) - ln (104)) - \frac{3}{2} (arctan (\frac{11}{2}) - arctan (5))

Simplify

\frac{1}{2} (3 ln (5) - ln (104)) - \frac{3}{2} (arctan (\frac{11}{2}) - arctan (5)) : \frac{3 ln ( 5 ) - ln ( 104 ) - 3 ( arctan ( \frac{11}{2} ) - arctan ( 5 ) )}{2}

= \frac{3 ln ( 5 ) - ln ( 104 ) - 3 ( arctan ( \frac{11}{2} ) - arctan ( 5 ) )}{2}

What is the integral from 7 to 8 of x/(x^2+6x+13) ?
The integral from 7 to 8 of x/(x^2+6x+13) is (3ln(5)-ln(104)-3(arctan(11/2)-arctan(5)))/2