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How do you use substitution to integrate # f(x)= x/sqrt(20+x^2)#?

Answer 1

Emma Aldridge

#int x/sqrt(20+x^2)dx = sqrt(20+x^2)+C#

Substitute #u = 20+x^2#, #du = 2xdx#

#int x/sqrt(20+x^2)dx = 1/2 int (2xdx)/sqrt(20+x^2) = int (du)/(2sqrtu) = sqrt(u)+C#

and undoing the substitution:

#int x/sqrt(20+x^2)dx = sqrt(20+x^2)+C#

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Answer 2

Charles Anctil

To integrate ( f(x) = \frac{x}{\sqrt{20 + x^2}} ) using substitution:

Let ( u = 20 + x^2 ).
Find ( \frac{du}{dx} ) and express ( dx ) in terms of ( du ).
Substitute ( u ) and the expression for ( dx ) into the integral.
Express ( x ) in terms of ( u ) within the integral.
Simplify the integral in terms of ( u ).
Integrate with respect to ( u ).
Finally, substitute back ( x ) for ( u ) to express the result in terms of ( x ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.