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Calculus/Improper Integrals

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Expert: - 2/25/2004


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

Answer
To find S (x^3)(cos(x^2)) dx ,
let u = x^2.

Then du/dx = 2x and dx = du/2x.

S(x^3)(cos(x^2)) dx = S x^3 cosu du/2x = S (x^2)/2 cosu du =

S u/2 cosu du =

(1/2) S u cosu du

To find S u cosu du , integrate by parts and get

S u cosu du =
u sinu - S sinu du =
u sinu + cos u

So (1/2) S u cosu du = (1/2)(u sinu + cos u)

Substitute u = x^2 and get

S (x^3)(cos(x^2)) dx = (1/2)((x^2)sin(x^2) + cos(x^2)) + C  

Calculus

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I can answer questions from the standard four semester Calculus sequence. I am not prepared for questions on Tensor Calculus. Everything else is welcome. Derivatives, partial derivatives, ordinary differential equations, single and multiple integrals, change of variable, vector integration (Green`s Theorem, Stokes, and Gauss) and applications.

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Ph.D. in Mathematics and many years teaching Calculus at state universities.

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