Evaluate the definite integral.
S b= sqrt(Pi) a= 0 xcos(x^2)dx
I'm not sure if this is right?
u= x^2
du= 2xdx
du/2= xdx
S (1/2)cos(u)
S (1/2)*sin(x^2)
[0.5 * sin(sqrt(Pi))^2] - [0.5 * sin(0)^2]
0 - 0 = 0, so zero is the answer?
S b= sqrt(Pi) a= 0 xcos(x^2)dx
What is "S b = sqrt(Pi) a= 0"
What are you integrating?
Is it this?
x cos(x^2) dx ?
from a = 0 to b = (sqrt(pi))?
S is supposed to be the integral symbol and b is the upper bound and a is the lower bound. Yes, xcos(x^2) dx is it.
| x cos(x^2) dx
| = integral symbol
By half-angle cos^2 x = 1/2 (1 + cos 2x)
| x 1/2 (1 + cos 2x) dx
1/2 | x + x cos 2x dx
1/2 | x dx + 1/2 | x cos 2x dx
1/4 x^2 + 1/2 | x cos 2x dx
Integrate | x cos 2x dx by Integration by Parts
u = x, dv = cos 2x dx
du = dx, v = 1/2 sin 2x
1/4 x^2 + 1/2 (1/2 x sin 2x - | 1/2 sin 2x dx )
1/4 x^2 + 1/4x sin 2x - 1/4 |sin 2x dx)
u = 2x
du = 2 dx
1/2 du = dx
1/4 x^2 + 1/4x sin 2x - 1/4 |1/2 sin u du
1/4 x^2 + 1/4x sin 2x - 1/8 | sin u du
1/4 x^2 + 1/4x sin 2x - 1/8 (-cos u)+ C
1/4 x^2 + 1/4x sin 2x + 1/8 cos u + C
1/4 x^2 + 1/4x sin 2x + 1/8 cos 2x + C
You can't do it like you did above because
'cos^2 x' does not mean just the 'x' is squared.
cos^2 x means (cos x)^2
I did not do the a to b part. check back if you need help with that part.
Good Luck!
| x cos(x^2) dx
The first line should be
| x cos^2 x
I forgot to change it.
But the x is the only thing being squared I thought because it is within parentheses and then multiplied by xcos dx?
After all that, I confused myself and did the wrong problem I think.
IF your problem is
|x cos(x^2) dx
just like your 2nd reply says, ignore the above.
I wrote down | x cos (x^2) but integrated
| x cos^2 x dx
So, your very first post IS correct if it is cos x^2 and not cos^2 x.
BIG difference in the integration!!
Well, now you know how to integrate x cos^2 x if you have to in the future.
To evaluate the definite integral ∫[b to a] xcos(x^2)dx, you correctly used the substitution u = x^2, which gives du = 2xdx. Dividing both sides by 2, you have du/2 = xdx.
Next, you rewrote the integral as ∫(1/2)cos(u) du, using the substitution. Here, you made a small mistake. The correct substitution is u = x^2, not u = (x^2)^2 = x^4. So when substituting, you have to ensure that you use the correct substitution.
Using the correct substitution, we have the integral as ∫(1/2)cos(u) du, where a and b are the limits of integration. The limits of integration also need to be changed accordingly.
Now, let's evaluate the integral:
∫(1/2)cos(u) du = (1/2)∫cos(u) du
The integral of cos(u) with respect to u is sin(u):
(1/2)∫cos(u) du = (1/2)sin(u) + C, where C is the constant of integration.
To find the definite integral from a to b, we substitute x back in for u:
(1/2)sin(u) + C = (1/2)sin(x^2) + C
Evaluating the definite integral from a (0) to b (sqrt(Pi)) yields:
[(1/2)sin(sqrt(Pi)^2)] - [(1/2)sin(0^2)]
= (1/2)sin(Pi) - (1/2)sin(0)
= (1/2)(0) - (1/2)(0)
= 0.
Therefore, the value of the definite integral ∫[sqrt(Pi) to 0] xcos(x^2)dx is indeed 0.
It's important to note that when making substitutions, choosing the correct substitution and being careful with limits of integration are crucial to arriving at the correct answer.