What is the integral from 0 to 6 of (2x)/((x-7)^2) ?

The integral from 0 to 6 of (2x)/((x-7)^2) is 2(-ln(7)+6)

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integral from 0 to 6 of (2x)/((x-7)^2)

\int_{0}^{6} \frac{2 x}{( x - 7 ) ^{2}} d x

2 (- ln (7) + 6)

Decimal

8.10817 \dots

Solution steps

\int_{0}^{6} \frac{2 x}{( x - 7 ) ^{2}} d x

Take the constant out:

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

= 2 \cdot \int_{0}^{6} \frac{x}{( x - 7 ) ^{2}} d x

Apply u-substitution

= 2 \cdot \int_{- 7}^{- 1} \frac{u + 7}{u ^{2}} d u

Expand

\frac{u + 7}{u ^{2}} : \frac{1}{u} + \frac{7}{u ^{2}}

= 2 \cdot \int_{- 7}^{- 1} \frac{1}{u} + \frac{7}{u ^{2}} d u

Apply the Sum Rule:

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

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

\int_{- 7}^{- 1} \frac{1}{u} d u = - ln (7)

\int_{- 7}^{- 1} \frac{7}{u ^{2}} d u = 6

= 2 (- ln (7) + 6)

What is the integral from 0 to 6 of (2x)/((x-7)^2) ?
The integral from 0 to 6 of (2x)/((x-7)^2) is 2(-ln(7)+6)