Show the power rule for integration

Question

ScanSolve AI Tutor · Accepted Answer

The power rule for integration states that for any real number $n \neq -1$ :

\int x^n dx = \frac{x^{n+1}}{n+1} + C

where $C$ is the constant of integration.

To show this rule, we can differentiate the result $\frac{x^{n+1}}{n+1} + C$ with respect to $x$ and demonstrate that it returns the original function $x^n$ .

Start with the proposed integral result.
Let $F(x) = \frac{x^{n+1}}{n+1} + C$ .
Differentiate $F(x)$ with respect to $x$ .
$\frac{d}{dx} \left( \frac{x^{n+1}}{n+1} + C \right)$
Apply the constant multiple rule and the power rule for differentiation, $\frac{d}{dx}(x^k) = kx^{k-1}$ .
The derivative of a constant $C$ is 0.
$\frac{d}{dx} \left( \frac{1}{n+1} x^{n+1} + C \right) = \frac{1}{n+1} \cdot \frac{d}{dx}(x^{n+1}) + \frac{d}{dx}(C)$ $= \frac{1}{n+1} \cdot (n+1)x^{(n+1)-1} + 0$
Simplify the expression.
$= \frac{1}{n+1} \cdot (n+1)x^n$ $= x^n$
Since the derivative of $\frac{x^{n+1}}{n+1} + C$ is $x^n$ , the power rule for integration is shown to be correct.

Show the power rule for integration

Step-by-step solution

Start with the proposed integral result.

Differentiate $F(x)$ with respect to $x$ .

Apply the constant multiple rule and the power rule for differentiation, $\frac{d}{dx}(x^k) = kx^{k-1}$ .

Simplify the expression.

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Show the power rule for integration

Step-by-step solution

Start with the proposed integral result.

Differentiate F(x)F(x)F(x) with respect to xxx.

Apply the constant multiple rule and the power rule for differentiation, ddx(xk)=kxk−1\frac{d}{dx}(x^k) = kx^{k-1}dxd​(xk)=kxk−1.

Simplify the expression.

Still stuck on this one?

More Mathematics Questions

Differentiate $F(x)$ with respect to $x$ .

Apply the constant multiple rule and the power rule for differentiation, $\frac{d}{dx}(x^k) = kx^{k-1}$ .