Pleas, how solve this integral ?
1 Answer
Jun 24, 2018
Use the substitutions
Explanation:
Given
I=int_4^7sqrt((x+2)/(x-3))dx
Apply the substitution
I=int_1^4sqrt((u+5)/u)du
Multiply numerator and denominator by
I=int_1^4(u+5)/sqrt(u^2+5u)dx
Rearrange:
I=1/2int_1^4(2u+5)/sqrt(u^2+5u)du+5/2int_1^4 1/sqrt(u^2+5u)du
Complete the square in the denominator:
I=[sqrt(u^2+5u)]_1^4+5int_1^4 1/sqrt((2u+5)^2-25)du
Apply the substitution
I=6-sqrt6+5/2intsecthetad theta
Integrate directly:
I=6-sqrt6+5/2[ln|5sectheta+5tantheta|]
Reverse the last substitution:
I=6-sqrt6+5/2[ln|(2u+5)+sqrt((2u+5)^2-25)|]_1^4
Insert the limits of integration:
I=6-sqrt6+5/2ln(25/(7+2sqrt6))