Question

Integrate the given function with the help of integration by parts, the given term is $\int {x \times {5^x}dx}$?

Answer 1

Hint : Integration by-parts means solving the term by assuming some other variables and it can be applied on one term or on two terms which is in product. If one term is only given in the question then we have to assume one as the other term, and then respectively we can solve the question.
Formulae Used: $\Rightarrow \int {udv = uv - \int {vdu\,} }$

Complete step-by-step answer :
The given term is $\int {x \times {5^x}dx}$
Here in the question it is asked for solving by by-parts method, which states for any two terms which are in product form it says:
$\Rightarrow \int {udv = uv - \int {vdu\,} }$
Using this formulae for our question we can assume the terms as per given formulae and then solve accordingly. After solving we get:

$$\Rightarrow for\,our\,question\,let\,u = x\, \\\ \Rightarrow \dfrac{{du}}{{dx}} = 1 \\\ \Rightarrow du = dx \\\ now \\\ \Rightarrow dv = {5^x}dx \\\ \Rightarrow \int {dv = \int {{5^x}dx} } \\\ \Rightarrow v = \dfrac{{{5^x}}}{{\ln 5}} \\\ we\,plug\,these\,values\,to\,see\,that \\\ \Rightarrow \int {x({5^x})dx = x\left( {\dfrac{{{5^x}}}{{\ln 5}}} \right) - \int {\dfrac{{{5^x}}}{{\ln 5}}dx} } \\\ \Rightarrow \int {x({5^x})dx = x\left( {\dfrac{{{5^x}}}{{\ln 5}}} \right) - \dfrac{1}{{\ln 5}}\int {{5^x}dx} } \\\ \Rightarrow \int {x({5^x})dx = \left( {\dfrac{{x \times {5^x}}}{{\ln 5}}} \right) - \dfrac{1}{{\ln 5}}\left( {\dfrac{{{5^x}}}{{\ln 5}}} \right) + C} \\\ \Rightarrow \int {x({5^x})dx = \dfrac{{x{{\left( 5 \right)}^x}}}{{\ln 5}} - \dfrac{{{5^x}}}{{{{\left( {\ln 5} \right)}^2}}}} \;$$

This is our required final solution of the term by using the by-parts method.

Note : The given question can be solved by by-parts only or you have to break or add certain more variables and constants to convert the question into some other form then by certain other techniques you can solve the question, but here by parts method is best suitable.
Integration is used for finding the area of the curve by using the summation rule, that is by assuming a very small unit area as a single unit then integrating it with the boundary conditions to get the final result. Here this question needs only simple integration and no boundary conditions are given to put on.