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Section 9
Integration by Partial Fractions: Nonrepeated Linear Factors
Make the given changes in the integrands of the indicated examples of this section, and then find the resulting fractions to be used in the integration. Do not integrate.In Example $2,$ change the numerator to $10-x$
Make the given changes in the integrands of the indicated examples of this section, and then find the resulting fractions to be used in the integration. Do not integrate.In Example $3,$ change the numerator to $x^{2}-12 x-10$
Write out the form of the partial fractions, similar to that shown in Eq. (I) of Example 2, that would be used to perform the indicated integrations. Do not evaluate the constants.$$\int \frac{3 x+2}{x^{2}+x} d x$$
Write out the form of the partial fractions, similar to that shown in Eq. (I) of Example 2, that would be used to perform the indicated integrations. Do not evaluate the constants.$$\int \frac{9-x}{x^{2}+2 x-3} d x$$
Write out the form of the partial fractions, similar to that shown in Eq. (I) of Example 2, that would be used to perform the indicated integrations. Do not evaluate the constants.$$\int \frac{x^{2}-6 x-8}{x^{3}-4 x} d x$$
Write out the form of the partial fractions, similar to that shown in Eq. (I) of Example 2, that would be used to perform the indicated integrations. Do not evaluate the constants.$$\int \frac{2 x^{2}-5 x-7}{x^{3}+2 x^{2}-x-2} d x$$
Integrate each of the given functions.$$\int \frac{x+3}{(x+1)(x+2)} d x$$
Integrate each of the given functions.$$\int \frac{x+2}{x(x+1)} d x$$
Integrate each of the given functions.$$\int \frac{8 d x}{x^{2}-4}$$
Integrate each of the given functions.$$\int \frac{p-9}{2 p^{2}-3 p+1} d p$$
Integrate each of the given functions.$$\int \frac{x^{2}+3}{x^{2}+3 x} d x$$
Integrate each of the given functions.$$\int \frac{x^{3}}{x^{2}+3 x+2} d x$$
Integrate each of the given functions.$$\int_{0}^{1} \frac{2 t+4}{3 t^{2}+5 t+2} d t$$
Integrate each of the given functions.$$\int_{1}^{3} \frac{x-1}{4 x^{2}+x} d x$$
Integrate each of the given functions.$$\int \frac{4 x^{2}-10}{x(x+1)(x-5)} d x$$
Integrate each of the given functions.$$\int \frac{4 x^{2}+21 x+6}{(x+2)(x-3)(x+4)} d x$$
Integrate each of the given functions.$$\int \frac{12 x^{2}-4 x-2}{4 x^{3}-x} d x$$
Integrate each of the given functions.$$\int_{1}^{2} \frac{x^{3}+7 x^{2}+9 x+2}{x\left(x^{2}+3 x+2\right)} d x$$
Integrate each of the given functions.$$\int_{2}^{3} \frac{d R}{R^{3}-R}$$
Integrate each of the given functions.$$\int \frac{2 x^{3}+x-1}{x^{3}+x^{2}-4 x-4} d x$$
Integrate each of the given functions.$$\int \frac{d V}{\left(V^{2}-4\right)\left(V^{2}-9\right)}$$
Integrate each of the given functions.$$\int \frac{x^{3}+2 x}{x^{2}+x-2} d x$$
Integrate each of the given functions.$$\int \frac{2 x d x}{x^{4}-3 x^{3}+2 x^{2}}$$
Integrate each of the given functions.$$\int \frac{e^{x} d x}{e^{2 x}+3 e^{x}+2}$$
Solve the given problems by integration.$$\text { Derive the general formula } \int \frac{d u}{u(a+b u)}=-\frac{1}{a} \ln \frac{a+b u}{u}+C$$
Solve the given problems by integration.$$\text { Derive the general formula } \int \frac{d u}{u^{2}-a^{2}}=\frac{1}{2 a} \ln \frac{u-a}{u+a}+C$$
Solve the given problems by integration.Integrate $\int \frac{d x}{x-\sqrt[3]{x}}$ by first letting $x=u^{3}$
Solve the given problems by integration.To integrate $\int \frac{x^{2} d x}{(x-2)(x+3)},$ explain why $A$ and $B$ cannot be$$\text { found if we let } \frac{x^{2}}{(x-2)(x+3)}=\frac{A}{x-2}+\frac{B}{x+3}$$
Solve the given problems by integration.$$\begin{aligned}&\text { Integrate: } \int \frac{\cos \theta}{\sin ^{2} \theta+2 \sin \theta-3} d \theta\\&\text { substitution }u=\sin \theta .)\end{aligned}$$
Solve the given problems by integration.Find the first-quadrant area bounded by $y=1 /\left(x^{3}+3 x^{2}+2 x\right)$ $x=1,$ and $x=3$
Solve the given problems by integration.Find the volume generated if the region of Exercise 30 is revolved about the $y$ -axis.
Solve the given problems by integration.Find the $x$ -coordinate of the centroid of a flat plate that covers the region bounded by $y\left(x^{2}-1\right)=1, y=0, x=2,$ and $x=4$
Solve the given problems by integration.Find an equation of the curve that passes through $(1,0),$ and the general expression for the slope is $(3 x+5) /\left(x^{2}+5 x\right)$
The current $i$ (in $\mathrm{A}$ ) as a function of the time $t$ (in $\mathrm{s}$ ) in a certain electric circuit is given by $i=(4 t+3) /\left(2 t^{2}+3 t+1\right) .$ Find the total charge that passes through a point during the first second.
Solve the given problems by integration.The force $F$ (in $\mathrm{N}$ ) applied by a stamping machine in making a certain computer part is $F=4 x /\left(x^{2}+3 x+2\right),$ where $x$ is the distance (in $\mathrm{cm}$ ) through which the force acts. Find the work done by the force from $x=0$ to $x=0.500 \mathrm{cm}$.
Solve the given problems by integration.Under specified conditions, the time $t$ (in min) required to form$x$ grams of a substance during a chemical reaction is given by $t=\int d x /[(4-x)(2-x)] .$ Find the equation relating $t$ and $x$ if $x=0$ g when $t=0$ min.