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kupik [55]
3 years ago
6

What is the antiderivative of 3x/((x-1)^2)

Mathematics
1 answer:
Maslowich3 years ago
4 0

Answer:

\int \:3\cdot \frac{x}{\left(x-1\right)^2}dx=3\left(\ln \left|x-1\right|-\frac{1}{x-1}\right)+C

Step-by-step explanation:

Given

\int \:\:3\cdot \frac{x}{\left(x-1\right)^2}dx

\mathrm{Take\:the\:constant\:out}:\quad \int a\cdot f\left(x\right)dx=a\cdot \int f\left(x\right)dx

=3\cdot \int \frac{x}{\left(x-1\right)^2}dx

\mathrm{Apply\:u-substitution:}\:u=x-1

=3\cdot \int \frac{u+1}{u^2}du

\mathrm{Expand}\:\frac{u+1}{u^2}:\quad \frac{1}{u}+\frac{1}{u^2}

=3\cdot \int \frac{1}{u}+\frac{1}{u^2}du

\mathrm{Apply\:the\:Sum\:Rule}:\quad \int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx

=3\left(\int \frac{1}{u}du+\int \frac{1}{u^2}du\right)

as

\int \frac{1}{u}du=\ln \left|u\right|     ∵ \mathrm{Use\:the\:common\:integral}:\quad \int \frac{1}{u}du=\ln \left(\left|u\right|\right)

\int \frac{1}{u^2}du=-\frac{1}{u}        ∵     \mathrm{Apply\:the\:Power\:Rule}:\quad \int x^adx=\frac{x^{a+1}}{a+1},\:\quad \:a\ne -1

so

=3\left(\ln \left|u\right|-\frac{1}{u}\right)

\mathrm{Substitute\:back}\:u=x-1

=3\left(\ln \left|x-1\right|-\frac{1}{x-1}\right)

\mathrm{Add\:a\:constant\:to\:the\:solution}

=3\left(\ln \left|x-1\right|-\frac{1}{x-1}\right)+C

Therefore,

\int \:3\cdot \frac{x}{\left(x-1\right)^2}dx=3\left(\ln \left|x-1\right|-\frac{1}{x-1}\right)+C

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Answer:

2

Step-by-step explanation:

3+4+(-5)=7+(-5)=7-5=2

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3 years ago
Which similarity statements are true? Check all that apply. △JKL ~ △KML △JKM ~ △JKL △JKM ~ △KML △JMK ~ △JKL △JMK ~ △KML
Aliun [14]
The answer:

according to the image, the main theorem concerning right triangle similarity is as follow:
the altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to the original triangle and each other, 

in our case, KL is the altitude, and by applying theorem, we get three triangles that are similar:
therefore:
<span>△JKL ~ △JKM
</span><span>△JKM ~ △JKM
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Answer:

Step-by-step explanation:

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If f(x) is an exponential function where f(-1.5) 26 and
mariarad [96]

Answer:

f(10) = 1147.25

Step-by-step explanation:

Given

f(-1.5) = 26

f(5.5) = 7

Required

f(10)

An exponential function is represented as:

f(x) = ab^x

f(-1.5) = 26 impleies that:

26 = ab^{-1.5} --- (1)

f(5.5) = 7 implies that

7 = ab^{5.5} --- (2)

Divide (2) by (1)

26/7 = ab^{-1.5}/ab^{5.5}

3.71429 = b^{-1.5+5.5}

3.71429 = b^{4}

Take 4th root

b = 1.39

Substitute b = 1.39 in 26 = ab^{-1.5}

26 = a * 1.39^{-1.5}

26 = a * 0.6102

Solve for (a)

a = 26/0.6102

a = 42.61

f(10) is calculated as:

f(10) = ab^{10}

f(10) = 42.61 * 1.39^{10}

f(10) = 1147.25

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2 years ago
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