What is the integral of x/(8x-7) ?

The integral of x/(8x-7) is 1/64 (8x-7+7ln|8x-7|)+C

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integral of x/(8x-7)

\int \frac{x}{8 x - 7} d x

\frac{1}{64} (8 x - 7 + 7 ln ∣ 8 x - 7 ∣) + C

Solution steps

\int \frac{x}{8 x - 7} d x

Apply u-substitution

: \int \frac{u + 7}{64 u} d u

= \int \frac{u + 7}{64 u} d u

Take the constant out:

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

= \frac{1}{64} \cdot \int \frac{u + 7}{u} d u

Expand

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

= \frac{1}{64} \cdot \int 1 + \frac{7}{u} 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

= \frac{1}{64} (\int 1 d u + \int \frac{7}{u} d u)

\int 1 d u = u

\int \frac{7}{u} d u = 7 ln ∣ u ∣

= \frac{1}{64} (u + 7 ln ∣ u ∣)

Substitute back

u = 8 x - 7

= \frac{1}{64} (8 x - 7 + 7 ln ∣ 8 x - 7 ∣)

Add a constant to the solution

= \frac{1}{64} (8 x - 7 + 7 ln ∣ 8 x - 7 ∣) + C

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\frac{d}{d y} (x sin (x y))

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What is the integral of x/(8x-7) ?
The integral of x/(8x-7) is 1/64 (8x-7+7ln|8x-7|)+C