Evaluate.
int_(pi)^(pi) (sin(x))^(3) (cos(x))^(3) dx
int_(pi)^(pi) (sin(x))^(3) (cos(x))^(3) dx
Omitting the π², we have
∫sin³(x)cos³(x)dx
=∫sin³(x)(1-sin²(x))cos(x)dx
=∫sin³(x)(1-sin²(x))dsin(x)
=sin^4(x)/4-sin^6(x)
no the answer is not that
it should be zero because its pi-pi
I have evaluated the indefinite integral.
If the limits are indeed π to π, yes, the answer is zero.
You have posted π^π, which I am not sure what it means, that is why I have indicated: "ignoring π^π".
It appeared to me that it is a constant equal to π², which doesn't make any sense.
Sorry that I have omitted the important term " +C " which indicates that it is an indefinite integral.
its okay, thank you for all your help i really appreciated it!
To evaluate the given definite integral, we will first rewrite it in a way that can be more easily solved.
The given integral is:
∫[π][π] (sin(x))^3 (cos(x))^3 dx
Since both sin(x) and cos(x) have powers of 3, we can use the trigonometric identity:
(sin(x))^3 (cos(x))^3 = (sin(x) cos(x))^3 (1 - (sin(x))^2) (1 - (cos(x))^2)
Using the identity sin(x) cos(x) = (1/2)sin(2x), we can substitute and simplify the integral further:
∫[π][π] 1/8 (sin(2x))^3 (1 - (sin(x))^2) (1 - (cos(x))^2) dx
Now, let's split this integral into three parts to evaluate them separately.
First, we evaluate the integral of (sin(2x))^3. To do this, we can use the substitution u = sin(2x) and obtain:
∫[π][π] (1/8) u^3 du
Integrating this with respect to u gives:
(1/32) u^4 + C = (1/32) (sin(2x))^4 + C
Next, we consider the integral of (1 - (sin(x))^2). We can rewrite this as:
∫[π][π] (1 - (sin(x))^2) dx = ∫[π][π] (cos(x))^2 dx
Using the identity (cos(x))^2 = (1/2) (1 + cos(2x)), we have:
∫[π][π] (1/2) (1 + cos(2x)) dx
Integrating this with respect to x gives:
(1/2) (x + (1/2)sin(2x)) + C
Finally, for the integral of (1 - (cos(x))^2), we can rewrite it as:
∫[π][π] (1 - (cos(x))^2) dx = ∫[π][π] (sin(x))^2 dx
Using the identity (sin(x))^2 = (1/2) (1 - cos(2x)), we have:
∫[π][π] (1/2) (1 - cos(2x)) dx
Integrating this with respect to x gives:
(1/2) (x - (1/2)sin(2x)) + C
Now, we can combine these results and evaluate the definite integral:
∫[π][π] (sin(x))^3 (cos(x))^3 dx = (1/32) (sin(2x))^4 + (1/2) (x + (1/2)sin(2x)) - (1/2) (x - (1/2)sin(2x)) + C
Since we are evaluating the definite integral from π to π, the first term (1/32) (sin(2x))^4 evaluates to 0. Also, the x terms cancel out in this interval, so we are left with:
∫[π][π] (sin(x))^3 (cos(x))^3 dx = -C
Note that the constant of integration C will depend on the context or any boundary conditions that may be specified in the problem.