Use a suitable change of variables to evaluate the following integral: summa x(10 - 10x)^5 dx

To evaluate the integral \( \int x(10 - 10x)^5 \, dx \) using a suitable substitution, follow these steps:

### Step 1: Choose the substitution
Let's choose the substitution:
\[
u = 10 - 10x
\]
Then, differentiate \( u \) with respect to \( x \):
\[
\frac{du}{dx} = -10
\]
Thus,
\[
du = -10 \, dx
\]
or equivalently,
\[
dx = -\frac{1}{10} \, du
\]

Now, express \( x \) in terms of \( u \):
\[
x = \frac{10 - u}{10} = 1 - \frac{u}{10}
\]

### Step 2: Substitute in the integral
Substitute \( u \) and \( dx \) into the integral:
\[
\int x(10 - 10x)^5 \, dx = \int \left(1 - \frac{u}{10}\right) u^5 \left(-\frac{1}{10}\right) \, du
\]

Simplify the integral:
\[
\int x(10 - 10x)^5 \, dx = -\frac{1}{10} \int \left(u^5 - \frac{u^6}{10}\right) \frac{1}{10} \, du = -\frac{1}{100} \int \left(u^5 - \frac{u^6}{10}\right) du
\]

### Step 3: Integrate with respect to \( u \)
Now, integrate each term with respect to \( u \):
\[
-\frac{1}{100} \int u^5 \, du + \frac{1}{1000} \int u^6 \, du
\]
\[
-\frac{1}{100} \left(\frac{u^6}{6}\right) + \frac{1}{1000} \left(\frac{u^7}{7}\right)
\]
\[
= -\frac{u^6}{600} + \frac{u^7}{7000} + C
\]

### Step 4: Substitute back \( u = 10 - 10x \)
Finally, substitute \( u = 10 - 10x \) back into the equation:
\[
-\frac{(10 - 10x)^6}{600} + \frac{(10 - 10x)^7}{7000} + C
\]

### Final Answer:
The indefinite integral is:
\[
\int x(10 - 10x)^5 \, dx = -\frac{(10 - 10x)^6}{600} + \frac{(10 - 10x)^7}{7000} + C
\]

Would you like more details, or do you have any questions?

Here are some related questions to consider:

1. How would you evaluate the integral \( \int x(5 - 5x)^4 \, dx \) using substitution?
2. What is the integral of \( \int (1 - 2x)(3 - 3x)^2 \, dx \)?
3. Can you use substitution to evaluate \( \int (4 - 4x)^3 \, dx \)?
4. How do you evaluate \( \int x^2(1 - x)^3 \, dx \) using substitution?
5. What happens if you use a different substitution, like \( u = x(10 - 10x) \), for this integral?

**Tip:** When using substitution, ensure the new variable simplifies the integral, turning a complex expression into one that is easier to integrate.

Learn how to evaluate the integral \( \int x(10 - 10x)^5 \, dx \) using the substitution method. Follow a detailed step-by-step solution with the chosen substitution \( u = 10 - 10x \), simplifying the integral process. Suitable for advanced college-level students and anyone studying calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation