integrate sin^7xdx
if
sin^6(x)sin(x)dx...(1)
then
let m=cos(x)
dm/dx=-sin(x)
dm=-sin(x)dx
input 1 into 2
(1-cos^2(x)^3)sin(x)dx
(1-u^2)^3-du
-(1-u^2)^3du
-[1-3u^2+3u^4-u^6)
[-1+3u^2-3u^4+u^7)
now just integrate and input your 'm'
thanks you
let sin^7x =(sin^2(x))^3sin(x) dx.
sin^6x= (sin^2(x))^3
integral of(sin^2(x))^3sinx
using identity sin^2x= 1-cos^2x
integral of (1-cos^2x)^3sinx dx
what if u=cosx
du=-sinx.
-integral(-u^6+3u^4-3u^2+1)du.
=-(-u^7/7+3u^5/5-u^3+u)+k
substituting for u give us the following
-(-cosx^7x/7+3cox^5/5-cos^3x+cosx)+k
sorry i made a typo i mean u not m
To integrate sin^7(x) with respect to x, we can use a combination of trigonometric identities and integration techniques. Here's the step-by-step process:
1. Start by using the identity sin^2(x) = 1/2 - 1/2*cos(2x) to rewrite sin^7(x) as sin^6(x) * sin(x).
sin^7(x) = (sin^2(x))^3 * sin(x) = [(1/2 - 1/2*cos(2x))^3] * sin(x)
2. Expand the cubic expression [(1/2 - 1/2*cos(2x))^3] using the binomial expansion.
[(1/2 - 1/2*cos(2x))^3] = (1/2)^3 - 3*(1/2)^2*(cos(2x)) + 3*(1/2)*(cos(2x))^2 - (cos(2x))^3
3. Now, we have the expression [(1/2 - 1/2*cos(2x))^3] * sin(x).
Next, we will integrate each term separately.
- The integral of (1/2)^3 with respect to x is (1/8) * x.
- The integral of 3*(1/2)^2*(cos(2x)) with respect to x can be simplified using the double-angle formula:
cos(2x) = 2*cos^2(x) - 1.
The integral becomes:
3*(1/2)^2 * [(2*cos^2(x) - 1)] = (3/8) * (2*cos^2(x) - 1).
- The integral of 3*(1/2)*(cos(2x))^2 with respect to x can be simplified using the half-angle formula:
cos^2(x) = (1/2)*(1 + cos(2x)).
The integral becomes:
3*(1/2) * [(1/2)*(1 + cos(2x))]^2 = (3/8) * [(1/2)*(1 + cos(2x))]^2.
- The integral of (cos(2x))^3 with respect to x can be simplified using the trigonometric identity:
cos^3(x) = (1/4)*(3*cos(x) + cos(3x)).
The integral becomes:
[(1/4)*(3*cos(x) + cos(3x))] = (1/4) * (3*cos(x) + cos(3x)).
4. Finally, integrate each term obtained in the above steps with respect to x:
Integral of sin^7(x) = (1/8) * x - (3/8) * (2*cos^2(x) - 1) + (3/8) * [(1/2)*(1 + cos(2x))]^2 + (1/4) * (3*cos(x) + cos(3x)) + C
where C is the constant of integration.
Note: The final expression may still be simplified further if desired, by expanding and simplifying the terms.