Wednesday

July 23, 2014

July 23, 2014

Posted by **Jason** on Thursday, September 22, 2011 at 2:23pm.

INT dx/(x^2*sqrt(x^2-25))

and

INT (from 0 to 1) -5*sqrt(x^2+1) dx

- Calculus -
**Steve**, Thursday, September 22, 2011 at 3:20pmRun, do not walk to the section on trig substitutions. Usually that's the best tool for problems like these.

Recall that sec^2 = 1 + tan^2, so that

sec^2 - 1 = tan^2. Let x=5sec(u), so x^2-25 = 25sec^2(u)-25 = 25 tan^2(u)

x^2 = 25 sec^2(u)

dx = 5 sec(u) tan(u)

Our integral now becomes

5sec(u)tan(u)/(25sec^2(u) * 5tan(u)) = 1/25sec(u) du = cos(u)/25 du

Int 1/25 cos(u) = 1/25 sin(u)

Since x = 5 sec(u),

x/5 = sec(u), and 5/x = cos(u)

sin(u) = sqrt(1-25/x^2) = 1/x * sqrt(x^2-25), and our answer is thus

1/25 sin(u) = sqrt(x^2-25)/25x

*whew*

The second one is similar, but uses hyperbolic functions:

Let x = tan(u), so x^2+1 = sec^2(u)

dx = sec^2(u) du

and the integral becomes

-5 Int sec^3(u) du

Use integration by parts (twice) to get

-5/2 tan(u)sec(u) - 5/2 log(tan u + sec u)

= -5/2 x sqrt(x^2+1) - 5/2 log(x + sqrt(x^2+1)

= -5/2[x sqrt(x^2+1) + arcsinh(x)]

from 0 to 1

At 0, the function is 0, so we just have f(1) = -5/2(sqrt(2) + asinh(1))

= -5/2(1.414 + 2.435) = 4.62

**Related Questions**

Calculus - Find arc length of y=logx from x=1 to x=2. dy/dx)^2=1/x^2 arc length=...

calculus - Find complete length of curve r=a sin^3(theta/3). I have gone thus- (...

MATHS - Find arc length of y=logx from x=1 to x=2. dy/dx)^2=1/x^2 arc length=Int...

Calculus - Graph the curve and find its exact length. x = e^t + e^-t, y = 5 - 2t...

Calculus AP - Volume of Revolution - Hi. I have a few calculus problems that I ...

Calc - f(x,y,z) = y; W is the region bounded by the plane x+y+z=2;cylinder x^2+z...

ap calculus - Which of the following definite integrals gives the length of y = ...

Calculus - Evaluate the indefinite integral: 8x-x^2. I got this but I the ...

Math Help please!! - Could someone show me how to solve these problems step by ...

calculus - Find circumference of the circle r=2acos theta. s= Int (0 to 2pi) of ...