Use partial fractions to solve the integral

\int \frac{x^2-x+20}{x^3+5x}dx

\int \frac{x^2-x+20}{x^3+5x}dx

beatricalwu
2022-07-16
Answered

Use partial fractions to solve the integral

\int \frac{x^2-x+20}{x^3+5x}dx

\int \frac{x^2-x+20}{x^3+5x}dx

You can still ask an expert for help

grocbyntza

Answered 2022-07-17
Author has **25** answers

$\int \frac{{x}^{2}-x+20}{{x}^{3}+5x}dx\phantom{\rule{0ex}{0ex}}\int \frac{{x}^{2}-x+20}{x({x}^{2}+5)}dx\phantom{\rule{0ex}{0ex}}\frac{A}{x}+\frac{Bx+C}{{x}^{2}+5}=\frac{{x}^{2}-x+20}{x({x}^{2}+5)}\phantom{\rule{0ex}{0ex}}A({x}^{2}+5)+Cx+5A={x}^{2}-x+20\phantom{\rule{0ex}{0ex}}A+B=1,C=-1,5A=20\phantom{\rule{0ex}{0ex}}B=-3,A=4\phantom{\rule{0ex}{0ex}}I=\int (\frac{4}{x}+\frac{-3x-1}{{x}^{2}+5})dx\phantom{\rule{0ex}{0ex}}=4\mathrm{ln}x-\frac{3}{2}\int \frac{2xdx}{{x}^{2}+5}-\int \frac{dx}{{x}^{2}+5}\phantom{\rule{0ex}{0ex}}=4\mathrm{ln}x-\frac{3}{2}\mathrm{ln}({x}^{2}+5)-\frac{1}{\sqrt{5}}{\mathrm{tan}}^{\prime}(\frac{x}{\sqrt{5}})+C$

$[\text{in the 2nd term let}{x}^{2}+5=v\text{}\text{}\text{}xdx=dx/2]\phantom{\rule{0ex}{0ex}}\int \frac{dx}{x}=\mathrm{ln}x+C\phantom{\rule{0ex}{0ex}}\int \frac{dx}{{x}^{2}+{a}^{2}}=\frac{1}{a}{\mathrm{tan}}^{-1}(\frac{x}{a})\phantom{\rule{0ex}{0ex}}I=4\mathrm{ln}x-\frac{3}{2}\mathrm{ln}({x}^{2}+5)-\frac{1}{\sqrt{5}}{\mathrm{tan}}^{-1}(\frac{x}{\sqrt{5}})+C$

$[\text{in the 2nd term let}{x}^{2}+5=v\text{}\text{}\text{}xdx=dx/2]\phantom{\rule{0ex}{0ex}}\int \frac{dx}{x}=\mathrm{ln}x+C\phantom{\rule{0ex}{0ex}}\int \frac{dx}{{x}^{2}+{a}^{2}}=\frac{1}{a}{\mathrm{tan}}^{-1}(\frac{x}{a})\phantom{\rule{0ex}{0ex}}I=4\mathrm{ln}x-\frac{3}{2}\mathrm{ln}({x}^{2}+5)-\frac{1}{\sqrt{5}}{\mathrm{tan}}^{-1}(\frac{x}{\sqrt{5}})+C$

asked 2022-05-22

Prove that $\frac{1}{1+{a}_{1}+{a}_{1}{a}_{2}}+\frac{1}{1+{a}_{2}+{a}_{2}{a}_{3}}+\cdots +\frac{1}{1+{a}_{n-1}+{a}_{n-1}{a}_{n}}+\frac{1}{1+{a}_{n}+{a}_{n}{a}_{1}}>1.$

I find it hard to use any inequalities here since we have to prove $>1$ and most inequalities such as AM-GM and Cauchy-Schwarz use $\ge 1$. On the other hand it seems that if I can prove that each fraction is $>1$ that might help, but I am unsure.

I find it hard to use any inequalities here since we have to prove $>1$ and most inequalities such as AM-GM and Cauchy-Schwarz use $\ge 1$. On the other hand it seems that if I can prove that each fraction is $>1$ that might help, but I am unsure.

asked 2022-07-16

How could I solve this chain rule problem?

To find: $\frac{d}{dt}[f(\mathbf{\text{c}}(t))]{|}_{t=0}$

Where $\mathbf{\text{c}}(t)$ is such that $\frac{d}{dt}\mathbf{\text{c}}(t)=\mathbf{\text{F}}(c(t))$, with $\mathbf{\text{F}}=\mathrm{\nabla}f$, $\mathbf{\text{c}}(0)=(-\frac{\pi}{2},\frac{\pi}{2})$ and $f(x,y)=\mathrm{sin}(x+y)xy$

I think I would need to proceed using the chain rule but I am currently not sure. How would I go about solving this problem?

Thanks in advance

To find: $\frac{d}{dt}[f(\mathbf{\text{c}}(t))]{|}_{t=0}$

Where $\mathbf{\text{c}}(t)$ is such that $\frac{d}{dt}\mathbf{\text{c}}(t)=\mathbf{\text{F}}(c(t))$, with $\mathbf{\text{F}}=\mathrm{\nabla}f$, $\mathbf{\text{c}}(0)=(-\frac{\pi}{2},\frac{\pi}{2})$ and $f(x,y)=\mathrm{sin}(x+y)xy$

I think I would need to proceed using the chain rule but I am currently not sure. How would I go about solving this problem?

Thanks in advance

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Find the quotient: $5\xf7\frac{1}{5}?$

asked 2022-04-01

Short expression $\frac{{x}^{2}-{y}^{2}}{x(x-y)}+\frac{{x}^{2}-{y}^{2}}{x(x+y)}$

I tried to short this expression:

$\frac{{x}^{2}-{y}^{2}}{x(x-y)}+\frac{{x}^{2}-{y}^{2}}{x(x+y)}$

The result should be 2 but I get:

$\frac{{x}^{2}-{y}^{2}}{{x}^{2}-xy}+\frac{{x}^{2}-{y}^{2}}{{x}^{2}+xy}$

$=\frac{{y}^{2}}{yx}+\frac{-{y}^{2}}{yx}$

$=0$

What did I do wrong?

I tried to short this expression:

The result should be 2 but I get:

What did I do wrong?

asked 2022-02-21

find the rational number halfway between the numbers 7/8 and 2/3

asked 2021-09-22

Discuss: Recognizing Partial Fraction Decompositions

For each expression, determine whether it is already a partial fraction decomposition or whether it can be decomposed further.

a)$\frac{x}{{x}^{2}+1}+\frac{1}{x+1}$

b)$\frac{x}{{(x+1)}^{2}}$

c)$\frac{1}{x+1}+\frac{2}{{(x+1)}^{2}}$

For each expression, determine whether it is already a partial fraction decomposition or whether it can be decomposed further.

a)

b)

c)

asked 2020-10-26

A taxi travels between two cities A and B, which are 100km apart. There are service stations at A and B and at the midpoint of the route. If the taxi breaks down, it does so at random at any point along the route between the cities. If a tow truck is dispatched from the nearest service station, what is the probability that it has to travel more than 15 km to reach the taxi