Q. \[ \int x^3 \, dx \]

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

\int x^3\,dx=\dfrac{x^4}{4}+C .

Q: How do you use the power rule for \int x^n\,dx ?

For n\neq -1 , \int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C . So with n=3 , \dfrac{x^4}{4}+C .

Q: Evaluate \int_{0}^{2} x^3\,dx .

Antiderivative: \dfrac{x^4}{4} . So \dfrac{2^4}{4}-\dfrac{0^4}{4}=\dfrac{16}{4}=4 .

Q: Check by differentiating \dfrac{x^4}{4}+C .

\dfrac{d}{dx}\left(\dfrac{x^4}{4}\right)=x^3 , since \dfrac{4x^3}{4}=x^3 .

Q: What is \int (x^3+5)\,dx ?

Split: \int x^3\,dx+\int 5\,dx=\dfrac{x^4}{4}+5x+C .

Answer

We integrate term by term using the power rule: for \(n \neq -1\), \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\).

\[
\int x^3 dx = \frac{x^{4}}{4} + C
\]

Final result: \(\frac{x^{4}}{4} + C\).

Detailed Explanation

We want to find the indefinite integral

\[
\int x^3\, dx
\]

Step 1: Use the power rule for integrals.

The power rule says that for any real number \(n \neq -1\),

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

Step 2: Identify \(n\) from the integrand.

Here, the integrand is \(x^3\), so \(n = 3\).

Step 3: Apply the rule by adding 1 to the exponent.

We compute \(n+1 = 3+1 = 4\), so the new power will be \(x^4\).

Step 4: Divide by the new exponent.

We divide by \(n+1 = 4\), giving \(\frac{x^4}{4}\).

Step 5: Add the constant of integration.

Because this is an indefinite integral, we add \(C\).

Final answer:

\[
\int x^3\, dx = \frac{x^4}{4} + C
\]

See full solution

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Calculus FAQ

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

\( \int x^3\,dx=\dfrac{x^4}{4}+C \).

How do you use the power rule for \( \int x^n\,dx \)?

For \( n\neq -1 \), \( \int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C \). So with \( n=3 \), \( \dfrac{x^4}{4}+C \).

Compute \( \int 3x^3\,dx \).

\( \int 3x^3\,dx=3\left(\dfrac{x^4}{4}\right)+C=\dfrac{3x^4}{4}+C \).

Evaluate \( \int_{0}^{2} x^3\,dx \).

Antiderivative: \( \dfrac{x^4}{4} \). So \( \dfrac{2^4}{4}-\dfrac{0^4}{4}=\dfrac{16}{4}=4 \).

Check by differentiating \( \dfrac{x^4}{4}+C \).

\( \dfrac{d}{dx}\left(\dfrac{x^4}{4}\right)=x^3 \), since \( \dfrac{4x^3}{4}=x^3 \).

What is \( \int (x^3+5)\,dx \)?

Split: \( \int x^3\,dx+\int 5\,dx=\dfrac{x^4}{4}+5x+C \).

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