Calculus Examples

$\int_{\sin (x)}^{1} \sqrt{1 + t^{2}} d t$

Step 1

Swap the bounds of integration.

$\frac{d}{d x} [- \int_{1}^{\sin (x)} \sqrt{1 + t^{2}} d t]$

Step 2

Take the derivative of $- \int_{1}^{\sin (x)} \sqrt{1 + t^{2}} d t$ with respect to $x$ using Fundamental Theorem of Calculus and the chain rule.

$\frac{d}{d x} [\sin (x)] (- \sqrt{1 + \sin^{2} (x)})$

Step 3

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$\cos (x) (- \sqrt{1 + \sin^{2} (x)})$

Step 4

Reorder the factors of $\cos (x) (- \sqrt{1 + \sin^{2} (x)})$ .

$- \sqrt{1 + \sin^{2} (x)} \cos (x)$