Question

$\int x(1-x)^{10}\,dx =$

• A. \(\frac{(1-x)^{12}}{12}-\frac{(1-x)^{11}}{11}+C\)
• B. \(\frac{(1-x)^{12}}{12}+\frac{(1-x)^{11}}{11}+C\)
• C. \(\frac{(1-x)^{11}}{11}-\frac{(1-x)^{10}}{10}+C\)
• D. \(\frac{(1-x)^{12}}{12}+\frac{(1+x)^{11}}{11}+C\)
• E. \(\frac{(1-x)^{12}}{12}-\frac{(1+x)^{11}}{11}+C\)

Hint:

In integrals containing expressions like \(x(1-x)^n\), substitution \(u=1-x\) often makes the integral very simple. After substitution, expand and integrate term by term.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The integral involves a product of a simple polynomial \( x \) and a more complex term \( (1-x)^{10} \). This structure is ideal for the method of integration by substitution, which simplifies the integrand.
Step 2: Key Formula or Approach:
We use the substitution method. A good choice for substitution is the expression inside the power.
1. Let \( u = 1-x \).
2. Express \( x \) and \( dx \) in terms of \( u \) and \( du \).
3. Substitute these into the integral and evaluate the new integral in terms of \( u \).
4. Substitute back to express the result in terms of \( x \).
Step 3: Detailed Explanation:
Let \( u = 1-x \).
From this substitution, we can express \( x \) as \( x = 1-u \).
Differentiating \( u \) with respect to \( x \), we get \( \frac{du}{dx} = -1 \), which implies \( dx = -du \).
Now substitute \( x \), \( (1-x) \), and \( dx \) in the integral:
\[ \int x(1-x)^{10} dx = \int (1-u)(u)^{10} (-du) \] \[ = -\int (u^{10} - u^{11}) du \] \[ = \int (u^{11} - u^{10}) du \] Now, integrate term by term using the power rule \( \int u^n du = \frac{u^{n+1}}{n+1} \):
\[ = \frac{u^{12}}{12} - \frac{u^{11}}{11} + C \] Finally, substitute back \( u = 1-x \) to get the answer in terms of \( x \):
\[ = \frac{(1-x)^{12}}{12} - \frac{(1-x)^{11}}{11} + C \] Let's double-check by differentiating the result: \[ \frac{d}{dx} \left( \frac{(1-x)^{12}}{12} - \frac{(1-x)^{11}}{11} \right) = \frac{12(1-x)^{11}(-1)}{12} - \frac{11(1-x)^{10}(-1)}{11} \] \[ = -(1-x)^{11} + (1-x)^{10} = (1-x)^{10} [-(1-x) + 1] = (1-x)^{10} [-1+x+1] = x(1-x)^{10} \] The derivative matches the original integrand, so the result is correct.
Step 4: Final Answer:
The result of the integration is \( \frac{(1-x)^{12}}{12} - \frac{(1-x)^{11}}{11} + C \). This corresponds to option (A).