Home > Calculus > Formal Definition of the Definite Integral

How do you find the definite integral of #2 / (4+x^2) dx# from #[0, 2]#?

Answer 1

Emilia Aguilar

#pi/4#

The integral of #a/(a^2+x^2)# with respect to #x# is #tan^(-1)(x/a)#

#therefore# the integral of#2/(4+x^2)=tan^(-1)(x/2)# where #a=2#

So, #int_0^2 2/(4+x^2) dx#

#=[tan^(-1)(x/2)]_0^2#

#=tan^(-1)1-tan^(-1)0#

#=pi/4-0#

#=pi/4#

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Scarlett Allison

To find the definite integral of ( \frac{2}{4+x^2} ) from 0 to 2, you can use the formula for integrating rational functions. First, perform a substitution: ( x = 2 \tan(\theta) ). This transforms the integral into a trigonometric form. Then, integrate with respect to ( \theta ) from ( \theta = 0 ) to ( \theta = \arctan(1) ). After integrating, convert back to the original variable ( x ). The result is approximately 0.7854.

By signing up, you agree to our Terms of Service and Privacy Policy

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.