Integral of $x^{-3}$

Q: What is the integral of x^(-3)?

The integral of x^-3 is x^-2-2 + C.

The integral of $x^{-3}$ is $\frac{x^{-2}}{-2} + C$ .

\int x^{-3}\,dx = \frac{x^{-2}}{-2} + C

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Graph of f(x) = x^(-3) with the area from 1 to 2 shaded. The shaded area equals the definite integral, about 0.375.

How to work it out

1
We want an antiderivative of f(x) = $x^{-3}$ .
$\int x^{-3}\,dx$
2
Method: power rule with a negative exponent.
3
That gives the antiderivative; add the constant C because any constant disappears when you differentiate:
$\int x^{-3}\,dx = \frac{x^{-2}}{-2} + C$
4
Check by differentiating the answer; you must get the original function back:
$\frac{d}{dx}\left[\frac{x^{-2}}{-2}\right] = \frac{4 x^{-3}}{4}$
5
As a definite integral, evaluate the antiderivative at both ends and subtract. Between 1 and 2:
$\int_{1}^{2} x^{-3}\,dx = \Big[\frac{x^{-2}}{-2}\Big]_{1}^{2} \approx 0.375$

LaTeX source of the formulas on this page

\[ \int x^{-3}\,dx = \frac{x^{-2}}{-2} + C \]
\[ \int x^{-3}\,dx \]
\[ \frac{d}{dx}\left[\frac{x^{-2}}{-2}\right] = \frac{4 x^{-3}}{4} \]
\[ \int_{1}^{2} x^{-3}\,dx = \Big[\frac{x^{-2}}{-2}\Big]_{1}^{2} \approx 0.375 \]

Definite integrals of the same function

Interval	Value
from 1 to 2	0.375
from 1 to 1.5	0.27777778

Questions

What is the integral of

x^{-3}

The integral of $x^{-3}$ is $\frac{x^{-2}}{-2} + C$ .

Why do we add + C?

Every function F(x) + C has the same derivative, so the antiderivative is only defined up to a constant.

What is the integral of

x^{-3}

from 1 to 2?

About 0.375.

Integral of x−3x^{-3}x−3

How to work it out

Definite integrals of the same function

Questions

More integrals

Integral of $x^{-3}$