20+20+??? a. 80 b. 20 c .40 d. 100
2 answers:
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Integration by Parts.
It's not immediately obvious but this numerator differentiates very nicely.
Notice that taking the derivative of the numerator actually creates a factor of our denominator.
Ah ha! So we've found a good choice for our u,
letting dv be everything else.
Applying Integration by parts
gives us,
Simplifying things a little bit before integrating again,
and integrating the last term,
looking for a common denominator, multiplying the first term by 2/2 and the second term by (2x+1)/(2x+1),
Combine the fractions together, factor out the exponential,
combine like-terms as a final step,
and include a constant of integration,
It's not immediately obvious but this numerator differentiates very nicely.
Notice that taking the derivative of the numerator actually creates a factor of our denominator.
Ah ha! So we've found a good choice for our u,
letting dv be everything else.
Applying Integration by parts
gives us,
Simplifying things a little bit before integrating again,
and integrating the last term,
looking for a common denominator, multiplying the first term by 2/2 and the second term by (2x+1)/(2x+1),
Combine the fractions together, factor out the exponential,
combine like-terms as a final step,
and include a constant of integration,