## The indefinite integral you have is:

âˆ« 9x(5x-7)^11 dx

To solve this using the u-substitution method, you correctly set u = (5x-7).

Now, we need to find dx in terms of du. Since u = (5x-7), we can rearrange this equation to solve for x:

x = (u + 7)/5

Differentiating both sides with respect to u, we get:

(dx/du) = 1/5

Now, we can substitute dx/du into our integral:

âˆ« 9x(5x-7)^11 dx = âˆ« 9(u+7)(u^11) (dx/du) du

Using (dx/du) = 1/5, we have:

= (9/5) âˆ« (u+7)(u^11) du

Expanding the expression inside the integral, we get:

= (9/5) âˆ« (u^12 + 7u^11) du

Now, we can integrate each term separately:

âˆ« u^12 du = (1/13)u^13 + C1

âˆ« 7u^11 du = (7/12)u^12 + C2

Where C1 and C2 are constants of integration.

Finally, substituting back u = (5x-7):

= (9/5) [(1/13)(5x-7)^13 + C1] + (9/5) [(7/12)(5x-7)^12 + C2]

= (9/65)(5x-7)^13 + (63/20)(5x-7)^12 + C

Where C is the constant of integration combining C1 and C2.

So, the indefinite integral is:

âˆ« 9x(5x-7)^11 dx = (9/65)(5x-7)^13 + (63/20)(5x-7)^12 + C

Hope that helps! Let me know if you have any further questions.