Solve Integral Problem: Evaluate $\displaystyle\int^{1}_{0}{\sqrt{x^2+1}}$

[imranq]

Homework Statement

Evaluate $\displaystyle\int^{1}_{0}{\sqrt{x^2+1}}$

Homework Equations

The Attempt at a Solution

By trigonometric substitution: $x = \tan{\theta} \rightarrow dx = \sec^2{\theta}\,d\theta$<br /> \[\int^{\frac{\pi}{4}}_{0}{\sec^2{x}\sqrt{\tan^2{\theta}+1}}\,d\theta = \int^{\frac{\pi}{4}}_{0}{\sec^3{\theta}}\,d\theta\]<br /> \[= \int^{\frac{\pi}{4}}_{0}{\sec{\theta}\tan^2{\theta}+\sec{\theta}}\,d\theta\]

This is where I get stuck

 

[tiny-tim]

\int^{\frac{\pi}{4}}_{0}{\sec^2{\theta}\sqrt{\tan^2{\theta}+1}}\,d\theta = \int^{\frac{\pi}{4}}_{0}{\sec^3{\theta}}\,d\theta\]<br /> \[= \int^{\frac{\pi}{4}}_{0}{\sec{\theta}\tan^2{\theta}+\sec{\theta}}\,d\theta\]

This is where I get stuck

Hi imranq!

(have a theta: θ and a squared: ² and a cubed: ³ )

Hint: (d/dθ)(secθ tanθ) = … ?