Pleas, how solve this integral ?

enter image source here

1 Answer
Jun 24, 2018

Use the substitutions x-3=u and 2u+5=5sectheta.

Explanation:

Given

I=int_4^7sqrt((x+2)/(x-3))dx

Apply the substitution x-3=u:

I=int_1^4sqrt((u+5)/u)du

Multiply numerator and denominator by sqrt(u+5):

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 2u+5=5sectheta:

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))