Comprehension 1
Let `I_(n,m)=intsin^(n)xcos^(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)`, vi) `I_(n+2,m-2)`
Suppose we want to establish a relation between `I_(n,m)` and `I_(n,m-2)`, then we set
`P(x)=sin^(n+1)xcos^(m-1)x`................(1)
In `I_(n,m)` and `I_(n,m-2)` the exponent of `cosx` is m and `m-2+1=m-1`. Now choose the exponent `m-1` of `cosx` in P(x). Similarly choose hte exponent of `sinx` for P(x).
Now, differentiating both sides of (1), we get
`P^(')(x) = (n+1)sin^(n)xcos^(m)X-(m-1)sin^(n+2)Xcos^(m-2)X`
`=(n+1)sin^(n)Xcos^(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)Xcos^(m-2)X+(m-1)sin^(n)Xcos^(m)X`
`=(n+m)sin^(n)Xcos^(m)X-(m-1)sin^(n)Xcos^(m-2)X`
Now, integrating both sides, we get
`sin^(n+1)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
Comprehension 1
Let `I_(n,m)=intsin^(n)xcos^(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)`, vi) `I_(n+2,m-2)`
Suppose we want to establish a relation between `I_(n,m)` and `I_(n,m-2)`, then we set
`P(x)=sin^(n+1)xcos^(m-1)x`................(1)
In `I_(n,m)` and `I_(n,m-2)` the exponent of `cosx` is m and `m-2+1=m-1`. Now choose the exponent `m-1` of `cosx` in P(x). Similarly choose hte exponent of `sinx` for P(x).
Now, differentiating both sides of (1), we get
`P^(')(x) = (n+1)sin^(n)xcos^(m)X-(m-1)sin^(n+2)Xcos^(m-2)X`
`=(n+1)sin^(n)Xcos^(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)Xcos^(m-2)X+(m-1)sin^(n)Xcos^(m)X`
`=(n+m)sin^(n)Xcos^(m)X-(m-1)sin^(n)Xcos^(m-2)X`
Now, integrating both sides, we get
`sin^(n+1)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
Let `I_(n,m)=intsin^(n)xcos^(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)`, vi) `I_(n+2,m-2)`
Suppose we want to establish a relation between `I_(n,m)` and `I_(n,m-2)`, then we set
`P(x)=sin^(n+1)xcos^(m-1)x`................(1)
In `I_(n,m)` and `I_(n,m-2)` the exponent of `cosx` is m and `m-2+1=m-1`. Now choose the exponent `m-1` of `cosx` in P(x). Similarly choose hte exponent of `sinx` for P(x).
Now, differentiating both sides of (1), we get
`P^(')(x) = (n+1)sin^(n)xcos^(m)X-(m-1)sin^(n+2)Xcos^(m-2)X`
`=(n+1)sin^(n)Xcos^(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)Xcos^(m-2)X+(m-1)sin^(n)Xcos^(m)X`
`=(n+m)sin^(n)Xcos^(m)X-(m-1)sin^(n)Xcos^(m-2)X`
Now, integrating both sides, we get
`sin^(n+1)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))`
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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
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)x cos^(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)x cos^(3)x-3I_(2","2))`
D
`I_(4","2)=(1)/(4)(-sin^(3)x cos^(3)x+2I_(2","2))`
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_(6","2) is
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_(6","2) is
A
`I_(4","2)=1/5 (sin^(3)xcos^(3)x+8 I_(6","2))`
B
`I_(4","2)=1/5 (- sin^(3)xcos^(3)x+8 I_(6","2))`
C
`I_(4","2)=1/5 (sin^(3)xcos^(3)x- 8 I_(6","2))`
D
`I_(4","2)=1/6 (sin^(3)xcos^(3)x+8 I_(6","2))`
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_(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
A
`I_(4","2)=1/3 (sin^(5)x cos^(3)x+8I_(4","4))`
B
`I_(4","2)=1/3 (- sin^(5)x cos^(3)x+8I_(4","4))`
C
`I_(4","2)=1/3 (sin^(5)x cos^(3)x- 8I_(4","4))`
D
`I_(4","2)=1/3 (sin^(5)x cos^(3)x+6I_(4","4))`
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