Evaluate the integral: I = integral (3x² - 2x + 1) (x³
Evaluate the integral: $$ I = \int\left(3x^2 - 2x + 1\right)\left(x^3 - x^2 + x - 9\right)^7 dx. $$
Explanation of Method
When the integrand is a power of some expression multiplied by exactly the derivative of that expression, the substitution $u=(\text{the expression})$ makes $du$ match the extra factor perfectly, reducing the whole integral to a single power-rule application — with no leftover constant to track.
- Substitute $u$ for the base of the power.
- Recognize that $du$ exactly matches the remaining factor.
- Integrate the resulting power of $u$.
- Substitute back and add the constant of integration.
Step 1 · Set up the substitution
To solve this integral, we observe that the expression inside the parentheses, $x^3-x^2+x-9$, suggests that we should use substitution. Let:
$$ u = x^3 - x^2 + x - 9. $$
Now, differentiate both sides with respect to $x$:
$$ \frac{du}{dx} = 3x^2 - 2x + 1. $$
Rearranging, we get:
$$ du = (3x^2-2x+1)\,dx. $$
Step 2 · Rewrite and integrate
Thus, the integral simplifies to:
$$ I = \int u^7\,du. $$
This is a straightforward power integral:
$$ I = \frac{u^8}{8} + C. $$
Step 3 · Substitute back
Finally, substituting back $u = x^3-x^2+x-9$, we obtain:
$$ I = \frac{\left(x^3-x^2+x-9\right)^8}{8} + C. $$
Summary Table
| Quantity | Value |
|---|---|
| Substitution | $u = x^3-x^2+x-9$ |
| $du$ | $(3x^2-2x+1)\,dx$ — matches the other factor exactly |
| Antiderivative | $u^8/8$ |
Additional Notes
- The integrand is defined for all real $x$, since it is a polynomial once expanded.
- This is the cleanest possible substitution case: no constant factor needs to be introduced or removed, since $3x^2-2x+1$ is precisely $\tfrac{du}{dx}$, not just proportional to it.
- The result can be checked by differentiating $\dfrac{(x^3-x^2+x-9)^8}{8}$ with the chain rule and confirming it returns $(3x^2-2x+1)(x^3-x^2+x-9)^7$.
Result
The solution to the integral is: