What is the integral of (50)/((x-1)(x^2+49)) ?

The integral of (50)/((x-1)(x^2+49)) is ln|x-1|-1/2 ln|x^2+49|-1/7 arctan(x/7)+C

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integral of (50)/((x-1)(x^2+49))

\int \frac{50}{( x - 1 ) ( x ^{2} + 49 )} d x

ln ∣ x - 1 ∣ - \frac{1}{2} ln x^{2} + 49 - \frac{1}{7} arctan (\frac{x}{7}) + C

Solution steps

\int \frac{50}{( x - 1 ) ( x ^{2} + 49 )} d x

Take the constant out:

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

= 50 \cdot \int \frac{1}{( x - 1 ) ( x ^{2} + 49 )} d x

Take the partial fraction of

\frac{1}{( x - 1 ) ( x ^{2} + 49 )} : \frac{1}{50 ( x - 1 )} + \frac{- x - 1}{50 ( x ^{2} + 49 )}

= 50 \cdot \int \frac{1}{50 ( x - 1 )} + \frac{- x - 1}{50 ( x ^{2} + 49 )} d x

Apply the Sum Rule:

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

= 50 (\int \frac{1}{50 ( x - 1 )} d x + \int \frac{- x - 1}{50 ( x ^{2} + 49 )} d x)

\int \frac{1}{50 ( x - 1 )} d x = \frac{1}{50} ln ∣ x - 1 ∣

\int \frac{- x - 1}{50 ( x ^{2} + 49 )} d x = \frac{1}{50} (- \frac{1}{2} ln x^{2} + 49 - \frac{1}{7} arctan (\frac{x}{7}))

= 50 (\frac{1}{50} ln ∣ x - 1 ∣ + \frac{1}{50} (- \frac{1}{2} ln x^{2} + 49 - \frac{1}{7} arctan (\frac{x}{7})))

Simplify

50 (\frac{1}{50} ln ∣ x - 1 ∣ + \frac{1}{50} (- \frac{1}{2} ln x^{2} + 49 - \frac{1}{7} arctan (\frac{x}{7}))) : ln ∣ x - 1 ∣ - \frac{1}{2} ln x^{2} + 49 - \frac{1}{7} arctan (\frac{x}{7})

= ln ∣ x - 1 ∣ - \frac{1}{2} ln x^{2} + 49 - \frac{1}{7} arctan (\frac{x}{7})

Add a constant to the solution

= ln ∣ x - 1 ∣ - \frac{1}{2} ln x^{2} + 49 - \frac{1}{7} arctan (\frac{x}{7}) + C

ln (e^{x^{2}})

\int \frac{x ^{2}}{x ^{2} - 4 x + 13} d x

\int \frac{e ^{4 x}}{2} d x

x \to 0 lim (x \cdot ln (x))

x \to 3 lim (4^{f (x)})

What is the integral of (50)/((x-1)(x^2+49)) ?
The integral of (50)/((x-1)(x^2+49)) is ln|x-1|-1/2 ln|x^2+49|-1/7 arctan(x/7)+C