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What is the integral of #int ( 1 / (25 + x^2) ) dx #?

Answer 1

Scarlett Allison

#int(1/(25+x^2))dx=1/5 tan^-1 (x/5) +C#

#int(1/(25+x^2))dx #

#dx/d(theta)=5tantheta#

#dx= 5sec^2theta *(d)theta#

#int(1/(25+25tan^2theta))* 5sec^2theta*(d)theta#

#int(1/(25(1+tan^2theta)))* 5sec^2theta*(d)theta#

#1+tantheta=sec^2theta#

#int(1/(25(sec^2theta)))* 5sec^2theta*(d)theta#

#int(1/5)*(d)theta+C#

#x=5tantheta# #x/5 = tantheta# #tan^-1(x/5)=theta#

Plug in:

#=1/5 tan^-1 (x/5) +C#

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Answer 2

Aurora Alvarado

The integral of ( \frac{1}{25 + x^2} ) with respect to ( x ) is ( \frac{1}{5} \arctan\left(\frac{x}{5}\right) + 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.