Q. \[ \int x^2 \, dx \]

Q: What is the indefinite integral of x^2?

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

Q: How do you use the power rule to integrate x^n?

\displaystyle \int x^n\,dx = \frac{x^{n+1}}{n+1}+C for n\neq -1.

Q: What is \int 3x^2\,dx?

\displaystyle \int 3x^2\,dx = 3\cdot\frac{x^3}{3}+C=x^3+C.

Answer

To find the integral of \(x^2\), use the power rule:

\[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \ne -1).
\]

Here \(n=2\):

\[
\int x^2 \, dx = \frac{x^{3}}{3} + C.
\]

Detailed Explanation

We want to find the indefinite integral of \(x^2\). In other words, compute

\[
\int x^2 \, dx
\]

Step 1: Identify the integrand.

The integrand is \(x^2\). This means we are integrating a power of \(x\).

Step 2: Use the power rule for integration.

The power rule says:

\[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{for } n \neq -1
\]

Here, \(n = 2\), and since \(2 \neq -1\), the rule applies.

Step 3: Substitute \(n = 2\) into the rule.

\[
\int x^2 \, dx = \frac{x^{2+1}}{2+1} + C
\]

Step 4: Simplify the exponent and denominator.

The exponent becomes \(2+1 = 3\), and the denominator becomes \(2+1 = 3\):

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

Final Answer:

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

See full solution

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Calculus FAQ

What is the indefinite integral of \(x^2\)?

\(\displaystyle \int x^2\,dx = \frac{x^3}{3}+C\).

What is the definite integral \(\int_{0}^{2} x^2\,dx\)?

\(\displaystyle \int_{0}^{2} x^2\,dx=\left[\frac{x^3}{3}\right]_{0}^{2}=\frac{8}{3}-0=\frac{8}{3}\).

How do you use the power rule to integrate \(x^n\)?

\(\displaystyle \int x^n\,dx = \frac{x^{n+1}}{n+1}+C\) for \(n\neq -1\).

Why is there a constant of integration \(C\)?

Because derivatives of constants are zero, all antiderivatives differ by \(C\). If \(F'(x)=x^2\), then \(F(x)+C\) also works.

What is \(\int 3x^2\,dx\)?

\(\displaystyle \int 3x^2\,dx = 3\cdot\frac{x^3}{3}+C=x^3+C\).

What is \(\int (x^2+5)\,dx\)?

\(\displaystyle \int (x^2+5)\,dx=\int x^2\,dx+\int 5\,dx=\frac{x^3}{3}+5x+C\).

Use this to solve ∫x² simply.
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