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2 years ago

If ( 3 y − 9 ) 2 + 7 = 43 , then what is(are) the values of y?

Mathematics

1 answer:

n200080 [17]2 years ago

4 0

Answer:

y = 5 and y = 1

Step-by-step explanation:

$\textsc {Subtract 7 from each side :}$

⇒ (3y - 2)² + 7 - 7 = 43 - 7

⇒ (3y - 2)² = 36

$\textsc {take the square root on each side :}$

⇒ √(3y - 9)² = √36

⇒ 3y - 9 = ±6

$\textsc {When equal to +6, add 9 on each side :}$

⇒ 3y - 9 + 9 = 6 + 9

⇒ 3y = 15

$\textsc {Divide by 3 on each side :}$

⇒ 3y/3 = 15/3

⇒ y = 5

$\textsc {When equal to -6, add 9 on each side :}$

⇒ 3y - 9 + 9 = -6 + 9

⇒ 3y = 3

$\textsc {Divide by 3 on each side :}$

⇒ 3y/3 = 3/3

⇒ y = 1

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3 years ago

First make a substitution and then use integration by parts to evaluate the integral. (Use C for the constant of integration.) x

e-lub [12.9K]

Answer:

$(\frac{x^{2}-25}{2})ln(5+x)-\frac{x^{2}}{4}+\frac{5x}{2}+C$

Step-by-step explanation:

Ok, so we start by setting the integral up. The integral we need to solve is:

$\int x ln(5+x)dx$

so according to the instructions of the problem, we need to start by using some substitution. The substitution will be done as follows:

U=5+x

du=dx

x=U-5

so when substituting the integral will look like this:

$\int (U-5) ln(U)dU$

now we can go ahead and integrate by parts, remember the integration by parts formula looks like this:

$\int (pq')=pq-\int qp'$

so we must define p, q, p' and q':

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and now we plug these into the formula:

$\int (U-5)lnUdU=(\frac{U^{2}}{2}-5U)lnU-\int \frac{\frac{U^{2}}{2}-5U}{U}dU$

Which simplifies to:

$\int (U-5)lnUdU=(\frac{U^{2}}{2}-5U)lnU-\int (\frac{U}{2}-5)dU$

Which solves to:

$\int (U-5)lnUdU=(\frac{U^{2}}{2}-5U)lnU-\frac{U^{2}}{4}+5U+C$

so we can substitute U back, so we get:

$\int xln(x+5)dU=(\frac{(x+5)^{2}}{2}-5(x+5))ln(x+5)-\frac{(x+5)^{2}}{4}+5(x+5)+C$

and now we can simplify:

$\int xln(x+5)dU=(\frac{x^{2}}{2}+5x+\frac{25}{2}-25-5x)ln(5+x)-\frac{x^{2}+10x+25}{4}+25+5x+C$

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