How to differentiate and integrate something with a square root?

Question

How do I solve these, I am not sure how.

1. integrate$$\int_0^2 2\sqrt{1+x^2} \,dx$$

2. differentiate $$2\sqrt{1+2t^2}$$

Answer 1

The first is a fairly standard "trigonometric substitution" integral. I presume you know that $sin^2(\theta)+ cos^2(\theta)= 1$. Divide both sides by $cos^2(\theta)$ to get $tan^2(\theta)+ 1= sec^2(\theta)$. That implies that, for something involving a square root of $1+ x^2$ we should try the substitution $x= tan(\theta)$. Then $dx= sec^2(\theta)d\theta$. When x= 1, $\theta= arctan(1)= \pi/4$ and when x= 2, $\theta= arctan(2)$. The integral becomes $2\int_{\pi/4}^{arctan(2)} sec^3(\theta)d\theta$.

The second does not ask you to integrate- it asks you to differentiate $2\sqrt{1+ 2t^2}= 2(1+ 2t^2)^{1/2}$. Let $u= 1+ 2t^2$ so that the function is $2u^{1/2}$ and use the "chain rule": $\frac{2u^{1/2}}{dt}= \frac{2u^{1/2}}{du}\frac{du}{dt}$. Can you differentiate $2u^{1/2}$ with respect to u and $u= 1+ 2t^2$ with respect to t?