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How do I find the antiderivative of #f(x)=(5x^2) / (x^2 + 1)#?

Answer 1

Isabella Ackerman

You start by manipulating your function: #f(x)=(5x^2)/(x^2+1)==5-5/(x^2+1)#

Now you can integrate:

#intf(x)dx=int(5-5/(x^2+1))dx=# #=int5dx-int5/(x^2+1)dx=#

#=5x-5arctan(x)+c#

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Answer 2

Samantha Adkison

To find the antiderivative of ( f(x) = \frac{5x^2}{x^2 + 1} ), you can use the method of partial fraction decomposition followed by integration. After decomposing ( f(x) ) into partial fractions, you can integrate each term separately. The antiderivative will be:

[ \int \frac{5x^2}{x^2 + 1} , dx = \int \left(5 - \frac{5}{x^2 + 1}\right) , dx ]

[ = 5x - 5\arctan(x) + C ]

Where ( C ) is the constant of integration.

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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.