1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
ryzh [129]
3 years ago
14

"find the reduction formula for the integral" sin^n(18x)

Mathematics
1 answer:
dexar [7]3 years ago
7 0
Let

I(n,a)=\displaystyle\int\sin^nax\,\mathrm dx
For demonstration on how to tackle this sort of problem, I'll only work through the case where n is odd. We can write

\displaystyle\int\sin^nax\,\mathrm dx=\int\sin^{n-2}ax\sin^2ax\,\mathrm dx=\int\sin^{n-2}ax(1-\cos^2ax)\,\mathrm dx
\implies I(n,a)=I(n-2,a)-\displaystyle\int\sin^{n-2}ax\cos^2ax\,\mathrm dx

For the remaining integral, we can integrate by parts, taking

u=\sin^{n-3}ax\implies\mathrm du=a(n-3)\sin^{n-4}ax\cos ax\,\mathrm dx\mathrm dv=\sin ax\cos^2ax\,\mathrm dx\implies v=-\dfrac1{3a}\cos^3ax

\implies\displaystyle\int\sin^{n-2}ax\cos^2ax\,\mathrm dx=-\dfrac1{3a}\sin^{n-3}ax\cos^3ax+\dfrac{a(n-3)}{3a}\int\sin^{n-4}ax\cos^4ax\,\mathrm dx

For this next integral, we rewrite the integrand

\sin^{n-4}ax\cos^4ax=\sin^{n-4}ax(1-\sin^2ax)^2=\sin^{n-4}ax-2\sin^{n-2}ax+\sin^nax
\implies\displaystyle\int\sin^{n-4}ax\cos^4ax\,\mathrm dx=I(n-4,a)-2I(n-2,a)+I(n,a)

So putting everything together, we found

I(n,a)=I(n-2,a)-\displaystyle\int\sin^{n-2}ax\cos^2ax\,\mathrm dx
I(n,a)=I(n-2,a)-\left(-\dfrac1{3a}\sin^{n-3}ax\cos^3ax+\dfrac{n-3}3\displaystyle\int\sin^{n-4}ax\cos^4ax\,\mathrm dx\right)
I(n,a)=I(n-2,a)-\dfrac{n-3}3\bigg(I(n-4,a)-2I(n-2,a)+I(n,a)\bigg)+\dfrac1{3a}\sin^{n-3}ax\cos^3ax
\dfrac n3I(n,a)=\dfrac{2n-3}3I(n-2,a)-\dfrac{n-3}3I(n-4,a)+\dfrac1{3a}\sin^{n-3}ax\cos^3ax

\implies I(n,a)=\dfrac{2n-3}nI(n-2,a)-\dfrac{n-3}nI(n-4,a)+\dfrac1{na}\sin^{n-3}ax\cos^3ax
You might be interested in
Pls help I need the answer
alexandr402 [8]

Answer:

The first choice.

8 0
2 years ago
Read 2 more answers
The table below shows the movie preferences of high school and college students:
Finger [1]
<span>C. She calculated the conditional relative frequency for comedy films that are liked by college students. The correct answer is 28%. hope this helps please mark me as brainliest

</span>
8 0
3 years ago
Read 2 more answers
Which best describes how to solve the equation for x.
prohojiy [21]

Answer:

B. multiply both sides by c.

Step-by-step explanation:

To solve the equation for x, you have to isolate the x variable on one side of the equation. To do this, do the opposite of any operation applied to x to remove other variables from the x's side of the equation:

x/c = d

multiply both sides of the equation by c to isolate x:

(x/c) * c = d * c

x = dc

Now x can be solved for.

Hope this helps :)

7 0
2 years ago
Subtract 1 + ( - 1 ) - ( - 3 ). Hint: Add the first two numbers, then subtract the -3 from the result of that addition
alekssr [168]
The correct answer would be 3.

1+(-1)-(-3)
1+(-1)=0

0-(-3)
0+3= 3
7 0
3 years ago
Danny had 20 minutes to do a three-problem quiz. He spent 9 1/4 minutes on question A and 5 4/5 minutes on question B. How much
Travka [436]

Answer:

4 19/20

Step-by-step explanation:

9 1/4 + 5 4/5 = 15 1/20

20- 15 1/20 = 4  19/20

6 0
2 years ago
Other questions:
  • How to find the least common denominator of 3 fractions?
    11·1 answer
  • STATE THE DOMAIN OF EACH OF THE FOLLOWING RELATIONS.
    15·1 answer
  • Cameron’s bacteria population is modeled by an equation. Deon models his bacteria population with a graph. Cameron says that on
    15·2 answers
  • The surface area of the prism is ______ square units. All measurements in the image below are in units. (Input whole number only
    11·1 answer
  • Write an equation in slope intercept form for the line with slope -1/5and y intercept -7
    13·2 answers
  • What is the range for this set of data? 7, 15, 12Mrs. Juarez graded ten English papers and recorded the scores. 92, 95, 100, 62,
    14·1 answer
  • If a circle has a diameter of 8cm, what is the radius of this circle?
    11·1 answer
  • I need help please!!
    14·1 answer
  • Will the point of the arrow in Figure Bhave the same angle measure as the point of the arrow in Figure A? Explain your answer.
    7·1 answer
  • Please help me and tell me step by step
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!