Evaluate the integral: I = integral x (x² - 13)^23 dx
Evaluate the integral: $$ I = \int x\left(x^2 - 13\right)^{23} \, dx. $$
Explanation of Method
Whenever the integrand contains $x$ multiplying a power of $x^2$ plus a constant, the substitution $u=x^2-13$ is natural: its derivative is $2x\,dx$, which exactly matches the extra factor of $x$ present in the integrand (up to a constant), collapsing the whole problem to a single power-rule integral in $u$ — no matter how large the exponent.
- Substitute $u$ for the quadratic expression inside the parentheses.
- Match $x\,dx$ to $\tfrac12du$ using the derivative of $u$.
- Integrate the resulting power of $u$ using the power rule.
- Substitute back and add the constant of integration.
Step 1 · Set up the substitution
To solve this integral, we use the method of substitution. Let us first observe that the expression inside the parentheses, $x^2-13$, suggests that we should choose a substitution based on this term. Let:
$$ u = x^2 - 13. $$
Now, compute the derivative of $u$ with respect to $x$:
$$ \frac{du}{dx} = 2x \quad \text{or} \quad du = 2x\,dx. $$
Thus, $x\,dx = \dfrac{du}{2}$.
Step 2 · Rewrite the integral
Substituting into the integral, we get:
$$ I = \int x\left(x^2-13\right)^{23}dx = \int \frac{1}{2}u^{23}\,du. $$
This simplifies to:
$$ I = \frac{1}{2}\int u^{23}\,du. $$
Step 3 · Integrate
The integral of $u^{23}$ is straightforward:
$$ \int u^{23}\,du = \frac{u^{24}}{24}. $$
Therefore, the solution becomes:
$$ I = \frac{1}{2}\cdot \frac{u^{24}}{24} + C = \frac{u^{24}}{48} + C. $$
Step 4 · Substitute back $u = x^2 - 13$
Finally, substituting back $u = x^2-13$, we obtain:
$$ I = \frac{(x^2-13)^{24}}{48} + C. $$
Summary Table
| Quantity | Value |
|---|---|
| Substitution | $u = x^2-13$ |
| $x\,dx$ in terms of $du$ | $\tfrac12 du$ |
| Antiderivative | $\tfrac{u^{24}}{48}$ |
Additional Notes
- The integrand is defined for all real $x$, since it is a polynomial once expanded.
- This substitution works cleanly because the extra factor of $x$ in the integrand exactly matches (up to the constant $2$) the derivative of $x^2-13$ — this "derivative present as a factor" pattern is the hallmark of when $u$-substitution applies directly.
- The result can be checked by differentiating $\dfrac{(x^2-13)^{24}}{48}$ with the chain rule and confirming it returns $x(x^2-13)^{23}$.
Result
The solution to the integral is: