The integral of x(6x+7)^8

1 answer:

8 0

Hello,

Let's assume t=6x+7
x=(t-7)/6
dt=6dx

$\int\limits{x(6x+7)^8} \, dx = \int\limits { \dfrac{t-7}{6} t^8} \, \dfrac{dt}{6} \\

= \dfrac{1}{36} \int\limits{ (t^{9}-7t^8) \, dt = \dfrac{1}{36} ( \dfrac{t^{10}}{10}- \dfrac{7t^9}{9} )+C\\

= \dfrac{(6x+7)^9*(54x-7)}{3240} +C$

You might be interested in

What is the following quotient?

seropon [69]

Answer: Choice B) $\frac{5\sqrt{11}+5\sqrt{3}}{8}\\\\$

==========================================

Work Shown:

$\frac{5}{\sqrt{11}-\sqrt{3}}\\\\\\\frac{5(\sqrt{11}+\sqrt{3})}{(\sqrt{11}-\sqrt{3})(\sqrt{11}+\sqrt{3})}\\\\\\\frac{5\sqrt{11}+5\sqrt{3}}{(\sqrt{11})^2-(\sqrt{3})^2}\\\\\\\frac{5\sqrt{11}+5\sqrt{3}}{11-3}\\\\\\\frac{5\sqrt{11}+5\sqrt{3}}{8}\\\\\\$

This shows why choice B is the answer.

---------------

Explanation:

In the second step, I multiplied top and bottom by $(\sqrt{11}+\sqrt{3})$
This is so we can apply the difference of squares rule in step 3. The difference of squares rule is (x-y)(x+y) = x^2-y^2.
In step 4, the square roots cancel out with the squaring operation. The two operations are inverses of one another, which is why they cancel.

So in general, if the denominator is $\sqrt{a}-\sqrt{b}$ and you want to rationalize the denominator, then you should multiply top and bottom by $\sqrt{a}+\sqrt{b}$ . The same applies in reverse as well.

This leads to the denominator becoming $(\sqrt{a}-\sqrt{b})(\sqrt{a}+\sqrt{b}) = (\sqrt{a})^2-(\sqrt{b})^2 = a-b$

Keep in mind that $a \ge 0$ and $b \ge 0$

4 0

3 years ago

A researcher wishes to examine the relationship between years of schooling completed and the number of pregnancies in young wome

Ber [7]

Answer:

1. When amount of schooling increases by one year, the number of pregnancies will decrease by 4.

Step-by-step explanation:

Regression analysis is a statistical technique which is used for forecasting. It determines the relationship between two variables. It determines the relationship of two or more dependent and independent variables. It is widely used in stats to find trend in the data. It helps to predict the values of dependent and independent variables. In the given question, there is regression equation given. X and Y are considered as dependent variables. When number of schooling increases by 1 year then number of pregnancies will decrease by 4

5 0

3 years ago

What is the logarithmic form of the equation e3x ≈ 3247?

lesya [120]

E^(3x) = 3247

3x = log 3247

x = [log 3247]/3

8 0

3 years ago

Find the coordinates of the midpoint of the segment given its endpoints F(2,-6) and G(-8,5)

Bess [88]

Step-by-step explanation:

see the pic fof the answer