SOLUTION: Please show me how to solve this integral: INT(sqrt(90^2+(x-60)^2)dx)

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Question 913173: Please show me how to solve this integral: INT(sqrt(90^2+(x-60)^2)dx)
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
We need to use a trigonometric substitution to integrate.
Let's set up a right triangle.

Then from the diagram,
H%5E2=90%5E2%2B%28x-60%29%5E2
H=sqrt%2890%5E2%2B%28x-60%29%5E2%29
cos%28theta%29=90%2Fsqrt%2890%5E2%2B%28x-60%29%5E2%29
sec%28theta%29=%28sqrt%2890%5E2%2B%28x-60%29%5E2%29%29%2F90
90sec%28theta%29=sqrt%2890%5E2%2B%28x-60%29%5E2%29
From the diagram,
tan%28theta%29=%28x-60%29%2F90
tan%28theta%29=x%2F90-2%2F3
Integrating both sides,
sec%5E2%28theta%29d%28theta%29=%281%2F90%29dx
dx=90sec%5E2%28theta%29d%28theta%29
So now substituting,


You can look up this integration,

Reverting back to the diagram,
tan%28theta%29=x%2F90-2%2F3
sec%28theta%29=%281%2F90%29%28sqrt%2890%5E2%2B%28x-60%29%5E2%29%29