What is the integral of 7/(x(x^2+25)) ?

The integral of 7/(x(x^2+25)) is 7(1/25 ln|x|-1/50 ln|x^2+25|)+C

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integral of 7/(x(x^2+25))

\int \frac{7}{x ( x ^{2} + 25 )} d x

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

Solution steps

\int \frac{7}{x ( x ^{2} + 25 )} d x

Take the constant out:

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

= 7 \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 )}

= 7 \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

= 7 (\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

= 7 (\frac{1}{25} ln ∣ x ∣ - \frac{1}{50} ln x^{2} + 25)

Add a constant to the solution

= 7 (\frac{1}{25} ln ∣ x ∣ - \frac{1}{50} ln x^{2} + 25) + C

\frac{d ^{2}}{d x ^{2}} (sin (x^{2}))

\int csc^{4} (x) cot^{2} (x) d x

x \to 2 lim (5 x^{3} + x)

\frac{d}{d x} (\frac{x sin ( x )}{2 + cos ( x )})

x \to 0 + lim (ln (1))

What is the integral of 7/(x(x^2+25)) ?
The integral of 7/(x(x^2+25)) is 7(1/25 ln|x|-1/50 ln|x^2+25|)+C