Calculus Examples

$\int_{0}^{\cot (x)} \frac{1}{1 + t^{2}} d t$

Step 1

Take the derivative of $\int_{0}^{\cot (x)} \frac{1}{1 + t^{2}} d t$ with respect to $x$ using Fundamental Theorem of Calculus and the chain rule.

$\frac{d}{d x} [\cot (x)] \frac{1}{1 + \cot^{2} (x)}$

Step 2

The derivative of $\cot (x)$ with respect to $x$ is $- \csc^{2} (x)$ .

$- \csc^{2} (x) \frac{1}{1 + \cot^{2} (x)}$

Step 3

Combine $\frac{1}{1 + \cot^{2} (x)}$ and $\csc^{2} (x)$ .

$- \frac{\csc^{2} (x)}{1 + \cot^{2} (x)}$

Step 4

Simplify.

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Apply pythagorean identity.

$- \frac{\csc^{2} (x)}{\csc^{2} (x)}$

Cancel the common factor of $\csc^{2} (x)$ .

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Cancel the common factor.

$- \frac{\csc^{2} (x)}{\csc^{2} (x)}$

Rewrite the expression.

$- 1 \cdot 1$

Multiply $- 1$ by $1$ .

$- 1$