W03 Stepwise

Due date: Thursday 1/29, 11:59pm

Partial fractions

01

01

Distinct linear factors

Compute the integral:

$$\int \frac{1}{(x+2)(x-3)}dx$$

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Solution

Solutions - 0060-01

(1) Write the partial fractions general form equation:

$$\frac{1}{(x+2)(x-3)}=\frac{A}{x+2}+\frac{B}{x-3}$$

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(2) Solve for constants.

Cross multiply:

$$\gg \gg 1=A(x-3)+B(x+2)$$

Plug in $x=3$, obtain $1=B\cdot 5$ so $B=1/5$.

Plug in $x=-2$, obtain $1=A(-5)$ so $A=-1/5$.

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(3) Integrate each term:

$$\begin{array}{cc}\int \frac{1}{(x+2)(x-3)}dx& \gg \gg \int \frac{-1/5}{x+2}dx+\int \frac{1/5}{x-3}dx\\ & \\ & \gg \gg -\frac{1}{5}\ln |x+2|+\frac{1}{5}\ln |x-3|+C\end{array}$$Link to original

02

02

Long division first

Compute the integral:

$$\int \frac{2x^3+3x^2+7x+4}{x+1}dx$$

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Solution

Solutions - 0060-02

$$\int \frac{2x^3+3x^2+7x+4}{x+1}dx$$

(1) Numerator degree is not smaller! Long division first:

$$\frac{2x^3+3x^2+7x+4}{x+1}\gg \gg 2x^2+x+6-\frac{2}{x+1}$$

Now this already has the form of a partial fraction decomposition, so we proceed directly to integration.

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(2) Integrate using power rule (with log):

$$\begin{array}{c}\int 2x^2+x+6-\frac{2}{x+1}dx\\ \\ \gg \gg \frac{2}{3}{x}^3+\frac{1}{2}{x}^2+6x-2\ln |x+1|+C\end{array}$$Link to original

03

03

Repeated factor

Compute the integral:

$$\int \frac{1}{x{(x-1)}^2}dx$$

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Solution

Solutions - 0060-03

(1) Write the partial fractions general form equation:

$$\frac{1}{x{(x-1)}^2}\gg \gg \frac{A}{x}+\frac{B}{x-1}+\frac{C}{{(x-1)}^2}$$

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(2) Solve for constants.

Cross multiply:

$$1=A{(x-1)}^2+Bx(x-1)+Cx$$

Plug in $x=1$, obtain $1=C$.

Plug in $x=0$, obtain $1=A$.

Plug in $x=2$, obtain:

$$\begin{array}{c}1=A+B\cdot 2+C\cdot 2\\ \\ \gg \gg 1=1+2B+2\gg \gg B=-1.\end{array}$$

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(3) Integrate each term:

$$\begin{array}{cc}\int \frac{1}{x{(x-1)}^2}dx& \gg \gg \int \frac{1}{x}+\frac{-1}{x-1}+\frac{1}{{(x-1)}^2}\\ & \\ & \gg \gg \ln |x|-\ln |x-1|-{(x-1)}^{-1}+C\end{array}$$

Optional simplification:

$$\gg \gg \ln \left|\frac{x}{x-1}\right|-\frac{1}{x-1}+C$$Link to original

Simpson’s Rule

04

01

Simpson’s Rule

The chart above shows a record of ambient temperatures measured each 15 minutes over 3 hours. Compute the approximate average temperature using Simpson’s Rule. You may use a calculator to resolve the arithmetic in your final expression.

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Solution

Solutions - 0070-01

(1) Recall the formula for the average value of $f$ over $a\le x\le b$:

$$f_{\mathrm{avg}}=\frac{1}{b-a}{\int }_a^{b}f(x)dx$$

Here $a=0$ and $b=180$:

$$\gg \gg \frac{1}{180}{\int }_0^{180}f(x)dx$$

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(3) Use $\mathrm{\Delta }x=15$ in Simpson’s Rule:

\begin{align*} \int_{0}^{180}f(x)dx \quad \approx\quad S_{12} &\quad=\quad \frac{15}{3}\Big(21+4(21.3)+2(21.5)+\cdots+4(21.3)+21.2)\Big) \end{align*} ParseError: Invalid delimiter 'undefined' after '\Big'

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(4) Plug into average value formula:

$$f_{\mathrm{avg}}\approx \frac{S_{12}}{180}\gg \gg \approx {21.2}^{\circ }$$Link to original