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How do you do definite integrals with substitution?

Answer 1

The only thing different from indefinite integral is that you will have to convert the original lower and upper limits into the new ones.

Let us evaluate the definite integral below.

#int_0^2 2xe^{x^2+1} dx#

Let #u=x^2+1#. #Rightarrow {du}/{dx}=2x Rightarrow du=2x du#

When #x=0#, #u=(0)^2+1=1# When #x=2#, #u=(2)^2+1=5#

If x goes from 0 to 2, then u goes from 1 to 5.

By Substitution,

#int_0^2 2xe^{x^2+1} dx=int_1^5 e^u du=[e^u]_1^5=e^5-e#

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Answer 2

Axel Alvarez

To evaluate definite integrals using substitution, follow these steps:

Choose an appropriate substitution, typically denoted by ( u ), that simplifies the integral.
Compute ( du ) (the differential of ( u )) in terms of ( dx ).
Rewrite the integral in terms of ( u ) using the substitution and replace ( dx ) with ( du ).
Integrate the new expression with respect to ( u ).
Substitute back the original variable, if necessary, to express the final result in terms of the original variable.
Evaluate the definite integral using the original limits of integration.

Ensure that you also adjust the limits of integration appropriately when you substitute in terms of ( u ).

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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.