# Find the indefinite integral of the function given by y=5x*e^2x?

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The integral of y = 5x*e^2x can be found by using integration by parts.

This gives that Int[ u dv ]= u*v - Int [ v du]

For Int[ 5x*e^2x dx]

take u = 5x => du = 5*dx

dv = e^2x dx => v = e^2x/2

Substitute in the formula given

Int[ 5x*e^2x dx] = 5x*e^2x / 2 - Int [ 5*e^2x / 2 dx]

=> 5x*e^2x / 2 - (5/4)e^2x + C

**The required integral is (5/2)*x*e^2x - (5/4)e^2x + C**

We'll integrate by parts. For this reason, we'll consider the formula:

Int udv = u*v - Int vdu (*)

We'll put u = 5x. (1)

We'll differentiate both sides:

du =5 dx (2)

We'll put dv = e^2x (3)

We'll integrate both sides:

Int dv = Int e^3x dx

v = e^2x/2 (4)

We'll substitute (1) , (2) , (3) and (4) in (*):

Int udv = 5x*e^2x/2 - 5Int (e^2x/2)dx

**Int (5x*e^2x)dx = 5x*e^2x/2 - 5*e^2x/4 + C**