Home
Class 12
MATHS
Let I(m","n)= int sin^(n)x cos^(m)x dx. ...

Let `I_(m","n)= int sin^(n)x cos^(m)x dx`. Then , we can relate `I_(n ","m)` with each of the following :
(i) `I_(n-2","m) " " (ii) I_(n+2","m)`
(iii) `I_(n","m-2) " " (iv) I_(n","m+2)`
(v) `I_(n-2","m+2)" " I_(n+2","m-2)`
Suppose we want to establish a relation between `I_(n","m)` and `I_(n","m-2)`, then we get
`P(x)=sin^(n+1)x cos^(m-1)x` ...(i)
In `I_(n","m)` and `I_(n","m-2)` the exponent of cos x in m and `m-2` respectively, the minimum of the two is m - 2, adding 1 to the minimum we get `m-2+1=m-1`. Now, choose the exponent of sin x for m - 1 of cos x in P(x). Similarly, choose the exponent of sin x for
`P(x)=(nH)sin^(n)x cos^(m)x-(m-1)sin^(n+2) x cos^(m-2)x`.
Now, differentiating both the sides of Eq. (i), we get
`=(n+1)sin^(n)x cos^(m)x-(m-1)sin^(n)x(1-cos^(2)x)cos^(m-2)x`
`=(n+1)sin^(n)x cos^(m)x-(m-1)sin^(n)x cos^(m-2)x+(m-1)sin^(n)x cos^(n)x`
`=(n+m)sin^(n)x cos^(m)x-(m-1)sin^(n)x cos^(m-2)x`
Now, integrating both the sides, we get
`sin^(n+1)x cos^(m-1)x=(n+m)I_(n","m)-(m-1)I_(n","m-2)`
Similarly, we can establish the other relations.
The relation between `I_(4","2)` and `I_(2","2)` is

A

`I_(4,2)=1/6 (-sin^(3)xcos^(3)x+3I_(2,2))`

B

`I_(4,2)=1/6(sin^(3)x cos^(3)x-3I_(2,2))`

C

`I_(4,2)=1/6(sin^(3)xcos^(3)x-3I_(2,2))`

D

`I_(4,2)=1/6(sin^(3)xcos^(3)x-3I_(2,2))`

Text Solution

Verified by Experts

The correct Answer is:
`
A
Doubtnut Promotions Banner Mobile Dark
|

Topper's Solved these Questions

  • INDEFINITE INTEGRATION

    RESONANCE ENGLISH|Exercise Exercise-3 Part I- JEE ADVANCED/ IIT-JEE PROBLEMS|8 Videos

  • INDEFINITE INTEGRATION

    RESONANCE ENGLISH|Exercise PART-II JEE MAIN|9 Videos

  • INDEFINITE INTEGRATION

    RESONANCE ENGLISH|Exercise PART III: ONE OR MORE THAN ONE OPTIONS CORRECT TYPE|19 Videos

  • FUNDAMENTAL OF MATHEMATICS

    RESONANCE ENGLISH|Exercise Exercise|135 Videos

  • MATRICES & DETERMINANT

    RESONANCE ENGLISH|Exercise HLP|34 Videos

Similar Questions

Explore conceptually related problems

Let I_(m","n)= int sin^(n)x cos^(m)x dx . Then , we can relate I_(n ","m) with each of the following : (i) I_(n-2","m) " " (ii) I_(n+2","m) (iii) I_(n","m-2) " " (iv) I_(n","m+2) (v) I_(n-2","m+2)" " I_(n+2","m-2) Suppose we want to establish a relation between I_(n","m) and I_(n","m-2) , then we get P(x)=sin^(n+1)x cos^(m-1)x ...(i) In I_(n","m) and I_(n","m-2) the exponent of cos x in m and m-2 respectively, the minimum of the two is m - 2, adding 1 to the minimum we get m-2+1=m-1 . Now, choose the exponent of sin x for m - 1 of cos x in P(x). Similarly, choose the exponent of sin x for P(x)=(nH)sin^(n)x cos^(m)x-(m-1)sin^(n+2) x cos^(m-2)x . Now, differentiating both the sides of Eq. (i), we get =(n+1)sin^(n)x cos^(m)x-(m-1)sin^(n)x(1-cos^(2)x)cos^(m-2)x =(n+1)sin^(n)x cos^(m)x-(m-1)sin^(n)x cos^(m-2)x+(m-1)sin^(n)x cos^(n)x =(n+m)sin^(n)x cos^(m)x-(m-1)sin^(n)x cos^(m-2)x Now, integrating both the sides, we get sin^(n+1)x cos^(m-1)x=(n+m)I_(n","m)-(m-1)I_(n","m-2) Similarly, we can establish the other relations. The relation I_(4","2) and I_(4","4) is

Let I_(n)=int_(0)^(pi//2) sin^(n)x dx, nin N . Then

Let I_(n) = int_(0)^(pi)(sin^(2)(nx))/(sin^(2)x)dx , n in N then

Let I_(n)=int_(0)^(pi//2) cos^(n)x cos nx dx . Then, I_(n):I_(n+1) is equal to

If I(m,n)=int_0^1x^(m-1)(1-x)^(n-1)dx , then

If I_(m,n)= int_(0)^(1) x^(m) (ln x)^(n) dx then I_(m,n) is also equal to

If in I_(2)Cl_(6) the number of covalent bonds and coordinate bonds are m and n respectively. Find the sum of m+n

If I_(m)=int (sin x+cos x)^(m)dx , then show that m l_(m)=(sin x+ cos x)^(m-1)*(sin x- cos x)+2(m-1) I_(m-2)

If I_(m,n)= int(sinx)^(m)(cosx)^(n) dx then prove that I_(m,n) = ((sinx)^(m+1)(cosx)^(n-1))/(m+n) +(n-1)/(m+n). I_(m,n-2)

Let I_(n)=int_(0)^(1)x^(n)sqrt(1-x^(2))dx. Then lim_(nrarroo)(I_(n))/(I_(n-2))=

Home
Profile