What is the integral of 2/(x(x^2+25)) ?

The integral of 2/(x(x^2+25)) is 2(1/25 ln|x|-1/50 ln|x^2+25|)+C

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integral of 2/(x(x^2+25))

\int \frac{2}{x ( x ^{2} + 25 )} d x

2 (\frac{1}{25} ln ∣ x ∣ - \frac{1}{50} ln x^{2} + 25) + C

Solution steps

\int \frac{2}{x ( x ^{2} + 25 )} d x

Take the constant out:

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

= 2 \cdot \int \frac{1}{x ( x ^{2} + 25 )} d x

Take the partial fraction of

\frac{1}{x ( x ^{2} + 25 )} : \frac{1}{25 x} - \frac{x}{25 ( x ^{2} + 25 )}

= 2 \cdot \int \frac{1}{25 x} - \frac{x}{25 ( x ^{2} + 25 )} 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

= 2 (\int \frac{1}{25 x} d x - \int \frac{x}{25 ( x ^{2} + 25 )} d x)

\int \frac{1}{25 x} d x = \frac{1}{25} ln ∣ x ∣

\int \frac{x}{25 ( x ^{2} + 25 )} d x = \frac{1}{50} ln x^{2} + 25

= 2 (\frac{1}{25} ln ∣ x ∣ - \frac{1}{50} ln x^{2} + 25)

Add a constant to the solution

= 2 (\frac{1}{25} ln ∣ x ∣ - \frac{1}{50} ln x^{2} + 25) + C

cos^{2} (x) sin (x) y^{^{'}} + cos^{3} (x) y = 2

\int_{0}^{3} 2 π x (3 - x) d x

\frac{\partial}{\partial x} (4 ln (x y) - (x + y)^{2} + 5)

f (x) = \frac{( 3 x + 2 )}{( 3 x - 2 )}

\int (x ln (x)) d x

What is the integral of 2/(x(x^2+25)) ?
The integral of 2/(x(x^2+25)) is 2(1/25 ln|x|-1/50 ln|x^2+25|)+C