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]
2 years ago
14

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

Mathematics
1 answer:
dexar [7]2 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
1 Micah paints birdhouses to sell at a fair. The table shows
seropon [69]
Result is that b=12p

5 0
3 years ago
Yurrrrrrrr......................​
baherus [9]

Answer:

1000000000000000000000+10000000000000000000000 :)

Step-by-step:

4 0
2 years ago
Read 2 more answers
Jacob traveled 171 miles in 3.8 hours. He wants to know how many miles he traveled in one hour, so he set up this problem:
Makovka662 [10]

The answer is 45


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

8 0
2 years ago
Read 2 more answers
4 - (-3) simplify the expression
mario62 [17]

Answer:

7

Step-by-step explanation:

4-(-3)=4+3 (negative times negative is positive)

4+3=7

Hope that helps!

6 0
3 years ago
Read 2 more answers
Jerry buys a pack of pencils that cost $3 David buys 4 sets of markers each set of marker also cost 3 what is the total cost of
Marat540 [252]
The total cost of marker is 12
7 0
3 years ago
Other questions:
  • Jeremy made 2 quarts of lemonade which expression can be use to find the number of fluid ounces of lemonade jerry made
    7·1 answer
  • Find the radius of a circle that has an area of 6.76 cm². Use it for pi.
    9·1 answer
  • What is the area of a parallelogram when the height is 10 and the base is 15
    7·2 answers
  • Need help / easy please
    5·1 answer
  • I’d love the answer ASAP please
    7·1 answer
  • Given the equation y = 4x - 3.A) If x increases by 1 unit, what is the corresponding change in y? B) If x decreases by 5 units,
    5·1 answer
  • Round 4358 to the nearest thousand
    11·2 answers
  • If you solve this for me I’ll love you forever (but seriously I need the answer ASAP)
    15·2 answers
  • How many grams are there in 4 kilograms
    15·1 answer
  • A plane travels 2,000 kilometers at a speed of 900 kilometers per hour (kph) with no wind. when a tailwind is present, the plane
    13·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!