- #1

vande060

- 186

- 0

## Homework Statement

I think I have a neat way of solving this bugger, but I'm not sure if it is a mathematically "legal" route. I dont know if there is a smarter way to solve this or not..

∫ √(x^2 +1) dx from 0 to 1

## The Attempt at a Solution

I = ∫ √(x^2 +1) dx

x= tanϑ

√(x^2 +1) = secϑ

dx = sec^2ϑ

x(0) = 0

x(1) = pi/4

-pi/2 < ϑ < pi/2

I = ∫ secϑsec^2ϑ dϑ

= ∫secϑ(1 + tan^2ϑ) dϑ

= ∫secϑ dϑ + ∫secϑtan^2ϑ dϑ

= ∫secϑ dϑ + ∫secϑ(1 + sec^2ϑ) dϑ

= ∫secϑ dϑ + ∫secϑ dϑ + ∫ secϑsec^2ϑ dϑ

= ∫secϑ dϑ + ∫secϑ dϑ + I

2I = 2∫secϑ dϑ

I = ln |secϑ + tanϑ| + C from 0 to pi/4

I = ln| (2/√2) + 1| - 0