What procedure should you use to fit each integrand to the basic integration rules? Do not integrate. (a) $\int \frac{2+x}{x^{2}+9} d x$, (b) $\int \cot ^{2} x d x$.

Solution

Verified#### a)

$\textbf{To solve}$ this integral we will apply linearity:

$\textbf{This menas}$:

$\begin{align*} \int{\frac{2+x}{x^2+9}\ dx}&=2\int{\frac{1}{x^2+9}\ dx} +\int{\frac{x}{x^2+9}\ dx}\\ \end{align*}$

$\textbf{Now}$, we can see that we have basic integrals, but since we aren't asked we won't solve that.

$\textbf{Thus}$, we have just:

$\boxed{\textcolor{#c34632}{\text{Applied linearity}}}$

#### b)

$\textbf{To solve}$ this integral we will apply the rule:

$\color{#c34632}{\cot{x}=\frac{\cos{x}}{\sin{x}}}$

$\textbf{This leads}$ to:

$\begin{align*} \int{\cot^2{x}\ dx}&=\int{\frac{\overbrace{\cos^2{x}}^{1-\sin^2{x}}}{\sin^2{x}}\ dx}\\ &=\int{\frac{1-\sin^2{x}}{\sin^2{x}}\ dx}\\ &=\int{\frac{1}{\sin^2{x}}\ dx}-\int{dx}\\ \end{align*}$

$\textbf{Now}$, we can just use basic rules for integrals ...

$\textbf{Thus}$, our $\textbf{final solution}$ is:

$\boxed{\textcolor{#c34632}{\text{Apply trigonometric formula for $\cot{x}$ and llinearity}}}$

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