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.