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Question: How do you integrate \(\left( {\dfrac{5}{{\sqrt x }}} \right)dx\)?...

How do you integrate (5x)dx\left( {\dfrac{5}{{\sqrt x }}} \right)dx?

Explanation

Solution

According to the question we have to find the integration of (5x)dx\left( {\dfrac{5}{{\sqrt x }}} \right)dxwhich is as mentioned in the question. So, to find the integration of the expression first of all we have to rearrange the terms of the expression given in the question which can be done with the help of the formula as mentioned below:

Formula used:
⇒1x=x−1...............(A)\Rightarrow \dfrac{1}{x} = {x^{ - 1}}...............(A)
Now, after rearranging the terms of the expression we have to take all the integers/integers which are given in the expression.
Now, to find the integration of the expression obtained we have to use the formula to determine the integration which is as mentioned below:

∫xndx=1n+1xn+1+C................(B)\int {{x^n}dx = \dfrac{1}{{n + 1}}{x^{n + 1}} + C................(B)}
Where, C is the constant term.

Complete step by step solution:
Step 1: First of all we have to rearrange the terms of the expression given in the question which can be done with the help of the formula (A) as mentioned in the solution hint. Hence,
=∫5x12dx =∫5x−12dx  = \int {\dfrac{5}{{{x^{\dfrac{1}{2}}}}}dx} \\\ = \int {5{x^{ - \dfrac{1}{2}}}dx} \\\
Step 2: Now, after rearranging the terms of the expression we have to take all the integers/integers which are given in the expression.
⇒5∫x−12dx\Rightarrow 5\int {{x^{ - \dfrac{1}{2}}}dx}
Step 3: Now, to find the integration of the expression obtained we have to use the formula (B) to determine the integration which is as mentioned in the solution hint. Hence,
=5(1−12)x−12+1= 5\left( {\dfrac{1}{{ - \dfrac{1}{2}}}} \right){x^{ - \dfrac{1}{2} + 1}}
Now, we just have to find the L.C.M of the terms to obtain the required integration,
=5×2x12 =10x+C  = 5 \times 2{x^{\dfrac{1}{2}}} \\\ = 10\sqrt x + C \\\

Final solution: Hence, with the help of the formulas (A) and (B) we have determined the integration of the given expression which is(5x)dx\left( {\dfrac{5}{{\sqrt x }}} \right)dx =10x+C = 10\sqrt x + C.

Note:
It is necessary that we have to rearrange the terms of the given integration in which a variable x is given in the form of square root with the help of the formula (A) as mentioned in the solution hint.
While finding the integration it is necessary that we have to take all the constant terms or integers outside the integration.