What is the integral of (2x^3+6x^2+5)/x ?

The integral of (2x^3+6x^2+5)/x is (2x^3)/3+3x^2+5ln|x|+C

Popular >

integral of (2x^3+6x^2+5)/x

\int \frac{2 x ^{3} + 6 x ^{2} + 5}{x} d x

\frac{2 x ^{3}}{3} + 3 x^{2} + 5 ln ∣ x ∣ + C

Solution steps

\int \frac{2 x ^{3} + 6 x ^{2} + 5}{x} d x

Expand

\frac{2 x ^{3} + 6 x ^{2} + 5}{x} : 2 x^{2} + 6 x + \frac{5}{x}

= \int 2 x^{2} + 6 x + \frac{5}{x} 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 2 x^{2} d x + \int 6 x d x + \int \frac{5}{x} d x

\int 2 x^{2} d x = \frac{2 x ^{3}}{3}

\int 6 x d x = 3 x^{2}

\int \frac{5}{x} d x = 5 ln ∣ x ∣

= \frac{2 x ^{3}}{3} + 3 x^{2} + 5 ln ∣ x ∣

Add a constant to the solution

= \frac{2 x ^{3}}{3} + 3 x^{2} + 5 ln ∣ x ∣ + C

\int x^{3} arctan (x^{2} - 1) d x

\int \frac{a cos ( x ) + b}{sin ^{2} ( x )} d x

\int \frac{x ^{5}}{x ^{3} - 1} d x

\int sec^{6} (θ) tan (θ) d θ

What is the integral of (2x^3+6x^2+5)/x ?
The integral of (2x^3+6x^2+5)/x is (2x^3)/3+3x^2+5ln|x|+C