How do you integrate #int (4x)/sqrt(x^2-14x+40)dx# using trigonometric substitution?
1 Answer
Jun 28, 2018
Use the substitution
Explanation:
Let
#I=int(4x)/sqrt(x^2-14x+40)dx#
Complete the square in the denominator:
#I=4intx/sqrt((x-7)^2-9)dx#
Apply the substitution
#I=4int(3sectheta+7)/(3tantheta)(3secthetatanthetad theta)#
Simplify:
#L=4int(3sec^2theta+7sectheta)d theta#
Integrate term by term:
#L=4(3tantheta+7ln|3sectheta+3tantheta|)+C#
Reverse the substitution:
#L=4sqrt((x-7)^2-9)+28ln|x-7+sqrt((x-7)^2-9)|+C#