Calculus - Anti-Derivatives

posted by .

How would you find the

Integral of (cos(x\8))^3, defined from

-(4 x pi)/3) to (4 x pi)/3)

  • Calculus - Anti-Derivatives -

    cos^3(x)dx = cos^2(x)cos(x)dx =

    [1-sin^2(x)]cos(x)dx =

    [1-sin^2(x)]d(sin(x))

  • Calculus - Anti-Derivatives -

    Is cos(x)8 supposed to be
    cos 8x, (cosx)^8 or 8 cosx?

  • Calculus - Anti-Derivatives -

    Its actually supposed to be (cos(x/8))^3.

    With what Anonymous answered, I see he left out the x/8, instead only using x. Would you, Drwls, mind using x/8?

  • Calculus - Anti-Derivatives -

    Sorry, I missed the \ mark. It is always better to use / for fractions when typing.

    Without messing with trig identities for cos (x/8), let's just substitute u for x/8. Then your integral becomes

    Integral of (cos u)^3
    = (1- sin^2 u) cos u du , from

    u = -(pi/6) to (pi/6)

    Now let sin u = v
    dv = cos u du
    and your integral becomes

    Integral of (1- v^2)dv ,
    from v = -1/2 to 1/2,
    since that is what v is when u = +/- pi/6

    Now it should be easy!
    The indefinite integral is v - v^3/3. Evaluate it at v = 1/2 and -1/2.

    My answer: 11/24 - (-11/24) = 11/12

    Don't trust my algebra. Check it yourself.

  • Calculus - Anti-Derivatives -

    I can follow everything except when you change the (4 x pi)/3 to pi/6, and change pi/6 into 1/2. Could you clear that up for me?

  • Calculus - Anti-Derivatives -

    the upper limit for example was

    x = 4 pi/3
    but
    u = x/8
    so the upper limit using u instead of x is
    u = (4 pi/3) /8
    or
    u = pi/6

    then it changes again to go from u to v
    v = sin u
    so at that same upper limit where x = 4 pi/3 and u = pi/6, we need to find v

    so
    v at this upper limit is:
    v = sin (pi/6) = sin 30 degrees = 1/2

  • Calculus - Anti-Derivatives -

    Oh. Thank you, Damon, and Drwls. That really helped me out.

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. calc

    d/dx integral from o to x of function cos(2*pi*x) du is first i do the integral and i find the derivative right. by the fundamental theorem of calculus, if there is an integral from o to x, don't i just plug the x in the function. …
  2. please help me calc. have test tom

    d/dx integral from o to x of function cos(2*pi*x) du is first i do the integral and i find the derivative right. by the fundamental theorem of calculus, if there is an integral from o to x, don't i just plug the x in the function. …
  3. Integral

    That's the same as the integral of sin^2 x dx. Use integration by parts. Let sin x = u and sin x dx = dv v = -cos x du = cos x dx The integral is u v - integral of v du = -sinx cosx + integral of cos^2 dx which can be rewritten integral …
  4. calculus

    Let f be a function that has derivatives of all orders for all real numbers. Assume f(0)=5, f'(0)=-3, f''(0)=1, and f'''(0)=4. Write the third-degree Taylor polynomial for h, where h(x) = integral of f(t)dt from 0 to x, about x=0 for …
  5. calc

    find the area between the x-axis and the graph of the given function over the given interval: y = sqrt(9-x^2) over [-3,3] you need to do integration from -3 to 3. First you find the anti-derivative when you find the anti-derivative …
  6. calculus

    Use Fundemental Theorem of Calculus part 1 to find the derivatives of the following: (i) F(x)=Integral ( top3, bottom x) e^-t^2 dt (ii) G(theta)=Integral ( top cos theta, bottom sin theta) 1/1-x^2 dx
  7. Calculus

    I have two questions, because I'm preparing for a math test on monday. 1. Use the fundamental theorem of calculus to find the derivative: (d/dt) the integral over [0, cos t] of (3/5-(u^2))du I have a feeling I will be able to find …
  8. Math-Derivitives

    if f(x)=x^3+5x, then f'(2)= if f(x)=cos(2x), then f'(pi/4) derivatives and anti derivatives please help
  9. Calculus

    F(x) = cos(x) • the integral from 2 to x² + 1 of e^(u² +5)du Find F'(x). When i did this, i got: -2xsin(x)e^((x²+1)² + 5) But my teacher got: -sin(x) • the integral from x² + 1 of e^(u² +5)du + 2xcos(x)e^((x²+1)² + 5) Do …
  10. Calculus

    Find the outside and inside functions of the following to find their derivatives: 1) sqrt(2x+9) 2) cos(cos(x)) 3) tan(x) I already know how to find their derivatives I'm just not exactly sure what parts of the chain rule equation would …

More Similar Questions