Question

small question regarding integral of 1/x^3(sqrtx^2-1)

going to leave this problem here overnite, if somebody could please get back to me that would be great, i will look at this tomorrow.

http://gyazo.com/5d15b024789812ccbd52414a3f7469b1


I pretty much know how to do it, except the last part when i have to convert from theta back to x.
look at the picture i posted, and i dont understand how the person got their final answer on there, could somebody explain? where do the costsint go?

Since x/1=secx, then cos=1/x and sin =sqrtx^2-1/x according to pyt. theorem.

so, the final answer should be((1/2) * (arcsec x + (1/x)(sqrtx^2-1/x)+ C.

but the pic i posted doesnt have that answer.

Answer 1

You mean:\[\int\limits\frac{ 1 }{x ^{3} }\sqrt{x ^{2}-1}\]

Answer 2

multiply top and bottom with 2 x

=> ∫ 2x dx / [2 x^4 √(x^2 - 1) ]

let x^2 - 1 = t^2

x^2 = 1 + t^2

2x dx = 2t dt

Answer 3

= (1/2) tan^-1√(x^2 - 1) + (1/2) [√(x^2 - 1) / x^2 ] + C