What is the integral from 1 to 6 of (x^2+5)/(7x-x^2) ?

The integral from 1 to 6 of (x^2+5)/(7x-x^2) is -5+59/7 ln(6)

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integral from 1 to 6 of (x^2+5)/(7x-x^2)

\int_{1}^{6} \frac{x ^{2} + 5}{7 x - x ^{2}} d x

- 5 + \frac{59}{7} ln (6)

Decimal

10.10197 \dots

Solution steps

\int_{1}^{6} \frac{x ^{2} + 5}{7 x - x ^{2}} d x

Factor

\frac{x ^{2} + 5}{7 x - x ^{2}}

= \int_{1}^{6} \frac{x ^{2} + 5}{- x ( x - 7 )} d x

Take the constant out:

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

= - \int_{1}^{6} \frac{x ^{2} + 5}{x ( x - 7 )} d x

Expand

\frac{x ^{2} + 5}{x ( x - 7 )} : \frac{x ^{2}}{x ( x - 7 )} + \frac{5}{x ( x - 7 )}

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}^{6} \frac{x ^{2}}{x ( x - 7 )} d x + \int_{1}^{6} \frac{5}{x ( x - 7 )} d x)

\int_{1}^{6} \frac{x ^{2}}{x ( x - 7 )} d x = 5 - 7 ln (6)

\int_{1}^{6} \frac{5}{x ( x - 7 )} d x = - \frac{10}{7} ln (6)

= - (5 - 7 ln (6) - \frac{10}{7} ln (6))

Simplify

- (5 - 7 ln (6) - \frac{10}{7} ln (6)) : - 5 + \frac{59}{7} ln (6)

= - 5 + \frac{59}{7} ln (6)

What is the integral from 1 to 6 of (x^2+5)/(7x-x^2) ?
The integral from 1 to 6 of (x^2+5)/(7x-x^2) is -5+59/7 ln(6)