The Integration by Parts Method: How to Integrate x * 5^x

[whatlifeforme]

Homework Statement

integrate by parts.

Integral: x * 5^x

Homework Equations

The Attempt at a Solution

i got to (1/ln5) * 5^x ;; and I'm not sure how to integrate further.

 

[SammyS]

Homework Statement

integrate by parts.

Integral: x * 5^x

Homework Equations

The Attempt at a Solution

i got to (1/ln5) * 5^x ;; and I'm not sure how to integrate further.

How about giving a few details regarding how you got that answer & where you are in the process of integration by parts.

 

[whatlifeforme]

integral (x * 5^x)

u=x; du=dx
dv=5^x ; v=(1/ln5)(5^x)

(x/ln5)5^x - integral ((1/ln5)(5^x) dx)

 

[Mathitalian]

Hi whatlifeforme :)

You have to use the formula:

$\int f(x)g'(x)dx = f(x)g(x)-\int f'(x)g(x)dx$

In this case

$f(x)= x\implies f'(x)= 1$

$g'(x)= 5^{x}= e^{x\ln(5)}\implies g(x)=\frac{e^{x\ln(5)}}{\ln(5)}= \frac{5^x}{\ln(5)}$


so $\int f(x)g'(x)dx = f(x)g(x)-\int f'(x)g(x)dx$

becomes

$\int x 5^x dx = x \frac{5^{x}}{\ln(5)}-\int \frac{5^{x}}{\ln(5)}dx$

Now you have to solve

$\int \frac{5^x}{\ln(5)}dx= \frac{1}{\ln(5)}\int 5^xdx$

;)

 

[whatlifeforme]

so would the final simplified answer be:

(x/ln5)(5^x) - (5^x/(ln(5)^2)

 

[SammyS]

so would the final simplified answer be:

(x/ln5)(5^x) - (5^x/(ln(5)^2)

... plus the constant of integration.

Yes.

Check it by differentiating.