What is the integral of (7(x^2+5))/(x(x^2+7)) ?

The integral of (7(x^2+5))/(x(x^2+7)) is ln|x^2+7|+5ln|x|+C

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integral of (7(x^2+5))/(x(x^2+7))

\int \frac{7 ( x ^{2} + 5 )}{x ( x ^{2} + 7 )} d x

ln x^{2} + 7 + 5 ln ∣ x ∣ + C

Solution steps

\int \frac{7 ( x ^{2} + 5 )}{x ( x ^{2} + 7 )} d x

Take the constant out:

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

= 7 \cdot \int \frac{x ^{2} + 5}{x ( x ^{2} + 7 )} d x

Expand

\frac{x ^{2} + 5}{x ( x ^{2} + 7 )} : \frac{x ^{2}}{x ( x ^{2} + 7 )} + \frac{5}{x ( x ^{2} + 7 )}

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{x ^{2}}{x ( x ^{2} + 7 )} d x + \int \frac{5}{x ( x ^{2} + 7 )} d x)

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

\int \frac{5}{x ( x ^{2} + 7 )} d x = 5 (\frac{1}{7} ln ∣ x ∣ - \frac{1}{14} ln x^{2} + 7)

= 7 (\frac{1}{2} ln x^{2} + 7 + 5 (\frac{1}{7} ln ∣ x ∣ - \frac{1}{14} ln x^{2} + 7))

Simplify

7 (\frac{1}{2} ln x^{2} + 7 + 5 (\frac{1}{7} ln ∣ x ∣ - \frac{1}{14} ln x^{2} + 7)) : ln x^{2} + 7 + 5 ln ∣ x ∣

= ln x^{2} + 7 + 5 ln ∣ x ∣

Add a constant to the solution

= ln x^{2} + 7 + 5 ln ∣ x ∣ + C

3 x^{3} - x^{2} - 10 x, - x^{2} + 2 x, [- 2, 2]

\int_{- 2}^{0} 2 x^{2} + 5 x + 6 d x

f (x) = - \frac{3}{2} x^{3} + \frac{2}{5} x^{2} - 4

g (t) = \frac{7}{8 t ^{5}}

\int 3 sin (6 x) d x

What is the integral of (7(x^2+5))/(x(x^2+7)) ?
The integral of (7(x^2+5))/(x(x^2+7)) is ln|x^2+7|+5ln|x|+C