integral of (2x^2+3)/x^3 — step-by-step solution

Let's integrate:

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

For

x

≠ 0, divide each numerator term by the power in the denominator:

\frac{2 x^{2} + 3}{x^{3}} = \frac{2}{x} + \frac{3}{x^{3}}

ⅰ

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

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

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

\int \frac{2}{x} d x = 2 \cdot \int \frac{1}{x} d x

We use

\int \frac{1}{x} d x = \ln | x |

. The absolute value covers both signs of the denominator, on any interval where

x \neq 0

ⓘ The derivative of

2 \cdot \ln | x |

\frac{2}{x}

, so

2 \cdot \ln | x |

is an antiderivative of

\frac{2}{x}

\int \frac{2}{x} d x

2 \cdot \ln | x |

■A constant factor stays in front of the integral. An antiderivative of x^n is x^(n+1)/(n+1), for n ≠ -1:

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

\int 3 x^{-3} d x

- \frac{3 x^{-2}}{2}

- \frac{3}{2 x^{2}}

2 \cdot \ln | x | - \frac{3}{2 x^{2}}

▶Indefinite integrals are defined up to an additive constant C, so this yields:

- \frac{3}{2 x^{2}} + 2 \cdot \ln | x | + C