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

⚠ 4x is read as the product 4*x: for a power, write 4^x

Let's integrate:

\int(4 x^{3} - \frac{3}{x^{4}})dx

We attempt to reverse an integration: starting from

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

and

\frac{d}{d x} (a^{n}) = n a^{(n - 1)}

, we differentiate using ^(m+1)

Let's perform the differentiation:

\frac{d}{d x} (x^{(- 4 + 1)})

\frac{d}{d x} (x^{-3})

We differentiate using:

\frac{d}{d x} (f^{n}) = n \cdot \frac{d}{d x} (f) \cdot f^{(n - 1)}

(where f =

x

)

-3 x^{-4}

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

You observe that the computed derivative and the original math expression are the same, except for a coefficient. So the integral is:

\frac{-3}{-3} \cdot \frac{1}{x^{3}} + \int(4 x^{3})dx

ℹ

\frac{-3}{-3}

\frac{3}{3}

‖

\int(4 x^{3})dx

4 \cdot \frac{x^{4}}{4}

\frac{3}{3} \cdot \frac{1}{x^{3}} + 4 \cdot \frac{x^{4}}{4}

ℹ

\frac{3}{3}

1

‖

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

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

‖

4 \cdot \frac{x^{4}}{4}

\frac{4 x^{4}}{4}

‖

\frac{4 x^{4}}{4}

x^{4}

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

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

x^{4} + \frac{1}{x^{3}} + C