What is the integral from 1 to 2 of ((x+2)/(x+7x^2)) ?

The integral from 1 to 2 of ((x+2)/(x+7x^2)) is (53ln(2)-13ln(15))/7

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integral from 1 to 2 of ((x+2)/(x+7x^2))

\int_{1}^{2} (\frac{x + 2}{x + 7 x ^{2}}) d x

\frac{53 ln ( 2 ) - 13 ln ( 15 )}{7}

Decimal

0.21887 \dots

Solution steps

\int_{1}^{2} \frac{x + 2}{x + 7 x ^{2}} d x

Expand

\frac{x + 2}{x + 7 x ^{2}} : \frac{x}{x + 7 x ^{2}} + \frac{2}{x + 7 x ^{2}}

Apply the Sum Rule:

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

= \int_{1}^{2} \frac{x}{x + 7 x ^{2}} d x + \int_{1}^{2} \frac{2}{x + 7 x ^{2}} d x

\int_{1}^{2} \frac{x}{x + 7 x ^{2}} d x = \frac{1}{7} (ln (15) - 3 ln (2))

\int_{1}^{2} \frac{2}{x + 7 x ^{2}} d x = 2 (4 ln (2) - ln (15))

= \frac{1}{7} (ln (15) - 3 ln (2)) + 2 (4 ln (2) - ln (15))

Simplify

\frac{1}{7} (ln (15) - 3 ln (2)) + 2 (4 ln (2) - ln (15)) : \frac{53 ln ( 2 ) - 13 ln ( 15 )}{7}

= \frac{53 ln ( 2 ) - 13 ln ( 15 )}{7}

What is the integral from 1 to 2 of ((x+2)/(x+7x^2)) ?
The integral from 1 to 2 of ((x+2)/(x+7x^2)) is (53ln(2)-13ln(15))/7