What is the integral of (12)/(x(x^2+25)) ?

The integral of (12)/(x(x^2+25)) is 12(1/25 ln|x|-1/50 ln|x^2+25|)+C

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integral of (12)/(x(x^2+25))

\int \frac{12}{x ( x ^{2} + 25 )} d x

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

Solution steps

\int \frac{12}{x ( x ^{2} + 25 )} d x

Take the constant out:

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

= 12 \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 )}

= 12 \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

= 12 (\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

= 12 (\frac{1}{25} ln ∣ x ∣ - \frac{1}{50} ln x^{2} + 25)

Add a constant to the solution

= 12 (\frac{1}{25} ln ∣ x ∣ - \frac{1}{50} ln x^{2} + 25) + C

csc^{2} (x) - 4

(x + 6 x y^{4}) d x + e^{x^{2}} y^{3} d y = 0

y^{^{''}} + 16 y = sec (4 x)

\frac{d}{d x} (x^{3})

What is the integral of (12)/(x(x^2+25)) ?
The integral of (12)/(x(x^2+25)) is 12(1/25 ln|x|-1/50 ln|x^2+25|)+C