What is the integral of (2x+7)/(x^2+2x-4) ?

The integral of (2x+7)/(x^2+2x-4) is

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integral of (2x+7)/(x^2+2x-4)

\int \frac{2 x + 7}{x ^{2} + 2 x - 4} d x

ln x^{2} + 2 x - 4 + \frac{1}{5} ln \frac{1}{5} (x + 1) + 1 - \frac{1}{5} ln \frac{1}{5} (x + 1) - 1 - \frac{7}{2 5} (ln \frac{1}{5} (x + 1) + 1 - ln \frac{1}{5} (x + 1) - 1) + C

Solution steps

\int \frac{2 x + 7}{x ^{2} + 2 x - 4} d x

Expand

\frac{2 x + 7}{x ^{2} + 2 x - 4} : \frac{2 x}{x ^{2} + 2 x - 4} + \frac{7}{x ^{2} + 2 x - 4}

Apply the Sum Rule:

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

= \int \frac{2 x}{x ^{2} + 2 x - 4} d x + \int \frac{7}{x ^{2} + 2 x - 4} d x

\int \frac{2 x}{x ^{2} + 2 x - 4} d x = ln x^{2} + 2 x - 4 + \frac{1}{5} ln \frac{1}{5} (x + 1) + 1 - \frac{1}{5} ln \frac{1}{5} (x + 1) - 1

\int \frac{7}{x ^{2} + 2 x - 4} d x = - \frac{7}{2 5} (ln \frac{x + 1}{5} + 1 - ln \frac{x + 1}{5} - 1)

= ln x^{2} + 2 x - 4 + \frac{1}{5} ln \frac{1}{5} (x + 1) + 1 - \frac{1}{5} ln \frac{1}{5} (x + 1) - 1 - \frac{7}{2 5} (ln \frac{x + 1}{5} + 1 - ln \frac{x + 1}{5} - 1)

Simplify

ln x^{2} + 2 x - 4 + \frac{1}{5} ln \frac{1}{5} (x + 1) + 1 - \frac{1}{5} ln \frac{1}{5} (x + 1) - 1 - \frac{7}{2 5} (ln \frac{x + 1}{5} + 1 - ln \frac{x + 1}{5} - 1) : ln x^{2} + 2 x - 4 + \frac{1}{5} ln \frac{1}{5} (x + 1) + 1 - \frac{1}{5} ln \frac{1}{5} (x + 1) - 1 - \frac{7}{2 5} (ln \frac{1}{5} (x + 1) + 1 - ln \frac{1}{5} (x + 1) - 1)

= ln x^{2} + 2 x - 4 + \frac{1}{5} ln \frac{1}{5} (x + 1) + 1 - \frac{1}{5} ln \frac{1}{5} (x + 1) - 1 - \frac{7}{2 5} (ln \frac{1}{5} (x + 1) + 1 - ln \frac{1}{5} (x + 1) - 1)

Add a constant to the solution

= ln x^{2} + 2 x - 4 + \frac{1}{5} ln \frac{1}{5} (x + 1) + 1 - \frac{1}{5} ln \frac{1}{5} (x + 1) - 1 - \frac{7}{2 5} (ln \frac{1}{5} (x + 1) + 1 - ln \frac{1}{5} (x + 1) - 1) + C

\frac{d}{d x} (\frac{cos ( x )}{1 + co tan ( x )})

\int 7 e^{x} + 2 sec^{2} (x) d x

\frac{d}{d x} (x (a + b x) e^{3 x})

\frac{d}{d x} (e^{- \frac{1}{x ^{3}}})

\int 6 x^{5} e^{7 x^{6}} d x

What is the integral of (2x+7)/(x^2+2x-4) ?
The integral of (2x+7)/(x^2+2x-4) is