Evaluate the integral: integral (5 - 2x)^9 dx
Evaluate the integral: $$ \int (5 - 2x)^9 \, dx. $$
Explanation of Method
When the integrand is a power of a linear expression in $x$, the substitution $u = (\text{linear expression})$ turns it into a simple power of $u$. Because the derivative of a linear expression is constant, $dx$ converts to $du$ times a constant factor, making the substitution especially clean.
- Substitute $u$ for the linear expression inside the power.
- Convert $dx$ into $du$ using the constant derivative.
- Integrate the resulting power of $u$.
- Substitute back and add the constant of integration.
Step 1 · Use substitution
To simplify the integral, we will use the substitution method. Let:
$$ u = 5 - 2x. $$
Then, the derivative of $u$ with respect to $x$ is:
$$ \frac{du}{dx} = -2 \quad \Rightarrow \quad dx = -\frac{du}{2}. $$
Substituting into the integral, we get:
$$ \int (5-2x)^9\,dx = \int u^9 \left(-\frac{du}{2}\right) = -\frac{1}{2}\int u^9\,du. $$
Step 2 · Integrate
Now, we integrate $u^9$:
$$ \int u^9\,du = \frac{u^{10}}{10}. $$
Thus, the integral becomes:
$$ -\frac{1}{2}\int u^9\,du = -\frac{1}{2}\cdot \frac{u^{10}}{10} = -\frac{1}{20}u^{10}. $$
Step 3 · Substitute back $u = 5 - 2x$
Now, we substitute back $u = 5-2x$ into the result:
$$ -\frac{1}{20}u^{10} = -\frac{1}{20}(5-2x)^{10}. $$
Step 4 · Add the constant of integration
Finally, we add the constant of integration $C$ to the result:
$$ \int (5-2x)^9\,dx = -\frac{1}{20}(5-2x)^{10} + C. $$
Summary Table
| Quantity | Value |
|---|---|
| Substitution | $u = 5-2x$ |
| $dx$ in terms of $du$ | $-\tfrac{1}{2}du$ |
| Antiderivative in $u$ | $-\tfrac{1}{20}u^{10}$ |
Additional Notes
- The integrand is defined for all real $x$, since it is a polynomial once expanded.
- The factor $-\tfrac{1}{2}$ appearing throughout comes directly from $du = -2\,dx$; forgetting this constant is a very common error in linear-substitution problems.
- The result can be checked by differentiating $-\tfrac{1}{20}(5-2x)^{10}$ with the chain rule and confirming it returns $(5-2x)^9$.
Result
The solution to the integral is: