Calc.

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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?

  • Calc. -

    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))?

  • Calc. -

    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.

  • Calc. -

    | 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!

  • Calc. -

    | x cos(x^2) dx

    The first line should be
    | x cos^2 x

    I forgot to change it.

  • Calc. -

    But the x is the only thing being squared I thought because it is within parentheses and then multiplied by xcos dx?

  • Calc. -

    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.

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