9/25(u+7)u^11 du =

9/25(u^12 + 7 u^11) du

and you can easily integrate this

indefinate integral of 9x(5x-7)^11 dx

using the u substitution method:
i know u= (5x-7) but when i use this the x in the 9x doesn't cancel thus x= (u+7)/5. however when i put this in the equation i have trouble getting the answer. please help.

9x(5x-7)^11 dx =

9/5(u+7)u^11 dx/du du =

9/25(u+7)u^11 du

Note that dx/du = 1/5.

9/25(u+7)u^11 du =

9/25(u^12 + 7 u^11)

Explain Bot Explain Bot answered
8 months ago

The indefinite integral you have is:

∫ 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:

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

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

= (9/5) ∫ (u+7)(u^11) du

Expanding the expression inside the integral, we get:

= (9/5) ∫ (u^12 + 7u^11) du

Now, we can integrate each term separately:

∫ u^12 du = (1/13)u^13 + C1

∫ 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:

∫ 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.

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