integral of (x^2+1)/(3x^4) — step-by-step solution

Let's compute the integral:

\int \frac{x^{2} + 1}{3 x^{4}} d x

For

x

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

\frac{x^{2} + 1}{3 x^{4}} = \frac{1}{3 x^{2}} + \frac{1}{3 x^{4}}

ⅰ

\int \frac{x^{2} + 1}{3 x^{4}} d x

\int \frac{1}{3 x^{2}} d x + \int \frac{1}{3 x^{4}} d x

\int \frac{1}{3 x^{2}} d x + \int \frac{1}{3 x^{4}} d 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{1}{3 x^{2}} d x

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

- \frac{x^{-1}}{3}

- \frac{1}{3 x}

\int \frac{1}{3 x^{4}} d x

\int \frac{x^{-4}}{3} d x

- \frac{x^{-3}}{9}

- \frac{1}{9 x^{3}}

- \frac{1}{3 x} - \frac{1}{9 x^{3}}

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

- \frac{1}{3 x} - \frac{1}{9 x^{3}} + C