integrate e7 x(e7 x + 3)5/2 dx the actual question is: ? ? ln 7 ln 5 e7 x(e7 x + 3)5/2 dx Solution We\'ll solve this integral using substitution technique: We\'ll substitute [e^(7x) + 3] by t. Differentiating both sides we\'ll get: [7e^(7x)dx = dt] [e^(7x)dx = dt/7] We\'ll re-write the integral: [int e^(7x)*(e^7x + 3)^(5/2)dx = int t^(5/2)dt/7] [int t^(5/2)dt/7 = t^(5/2 + 1)/(7(5/2 + 1)) + C] [int t^(5/2)dt/7 = t^(7/2)/(7*(7/2)) + C] [int e^(7x)*(e^7x + 3)^(5/2)dx = 2(e^(7x) + 3)^(7/2)/49 + C] Therefore, the requested integral of the given function is [int e^(7x)*(e^(7x)+3)^(5/2)dx = (2sqrt[(e^(7x) + 3)^7])/49 + C.].

1. integrate e7 x(e7 x + 3)5/2 dx
the actual question is:
?
? ln 7
ln 5
e7 x(e7 x + 3)5/2 dx
Solution
We'll solve this integral using substitution technique:
We'll substitute [e^(7x) + 3] by t. Differentiating both sides we'll get:
[7e^(7x)dx = dt]
[e^(7x)dx = dt/7]
We'll re-write the integral:
[int e^(7x)*(e^7x + 3)^(5/2)dx = int t^(5/2)dt/7]
[int t^(5/2)dt/7 = t^(5/2 + 1)/(7(5/2 + 1)) + C]
[int t^(5/2)dt/7 = t^(7/2)/(7*(7/2)) + C]
[int e^(7x)*(e^7x + 3)^(5/2)dx = 2(e^(7x) + 3)^(7/2)/49 + C]
Therefore, the requested integral of the given function is [int e^(7x)*(e^(7x)+3)^(5/2)dx =
(2sqrt[(e^(7x) + 3)^7])/49 + C.]