What is the integral of (2x+21)/(2x^2+9x-5) ?

The integral of (2x+21)/(2x^2+9x-5) is 2ln|2x-1|-ln|x+5|+C

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integral of (2x+21)/(2x^2+9x-5)

\int \frac{2 x + 21}{2 x ^{2} + 9 x - 5} d x

2 ln ∣ 2 x - 1 ∣ - ln ∣ x + 5 ∣ + C

Solution steps

\int \frac{2 x + 21}{2 x ^{2} + 9 x - 5} d x

Take the partial fraction of

\frac{2 x + 21}{2 x ^{2} + 9 x - 5} : \frac{4}{2 x - 1} - \frac{1}{x + 5}

= \int \frac{4}{2 x - 1} - \frac{1}{x + 5} 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

= \int \frac{4}{2 x - 1} d x - \int \frac{1}{x + 5} d x

\int \frac{4}{2 x - 1} d x = 2 ln ∣ 2 x - 1 ∣

\int \frac{1}{x + 5} d x = ln ∣ x + 5 ∣

= 2 ln ∣ 2 x - 1 ∣ - ln ∣ x + 5 ∣

Add a constant to the solution

= 2 ln ∣ 2 x - 1 ∣ - ln ∣ x + 5 ∣ + C

x \to - 1 lim ((x + 2 x^{3})^{4})

\int \frac{x ^{5}}{1 - 2 x ^{2}} d x

\int 1 + 4 r^{2} r d r

\int \frac{3}{e ^{x}} d x

f (x) = 2^{x^{2}}

What is the integral of (2x+21)/(2x^2+9x-5) ?
The integral of (2x+21)/(2x^2+9x-5) is 2ln|2x-1|-ln|x+5|+C