Help!!! Calculus Integral Question

I have to solve the following equation,

a. Integral if (x)/ (1 + x^4) dx

b. Integral from 1/6 to 1/2 of (csc (pi*t) * cot (pi*t)) dt

I have to do u- substitution and we have not covered this in class

"I have to do u- substitution and we have not covered this in class" Not a big deal that you did not do it in class, just google it and you can find plenty of tutorials and even videos, it is not something that requires classroom instruction.

(a) You can tell immediately that the result is (1/2) tan^{-1}(x^2) if you are familiar with the derivative of the inverse tangent function. Anyway,

let x^2 = tan(u), 2x dx = sec^2(u) du ---> x dx = (1/2) sec^2(u)du, and the denominator term becomes 1 + x^4 = 1 + tan^2(u) = sec^2(u) as per identity. Thus, we have

x dx / (1 + x^4)

(1/2) sec^2(u) du / sec^2(u)

= (1/2) du

= (1/2) u + C upon integrating

recall x^2 = tan(u), then u = tan^{-1}(x^2), so the answer is

= (1/2)tan^{-1}(x^2) + C...............[Ans.]

(b) csc(pi t) cot(pi t) dt


there is no need for substitution here, it is well-known that the derivative of csc( pi t) = - pi csc(pi t) cot(pi t), this is a derivative people just memorize (like derivatives of sine, cosine, tangent, secant, etc.). So by inspection we can see the integral of

integral{ csc(pi t) cot(pi t) dt}

= integral{ d/dt ( (-1/pi) cot(pi t))dt}

= - csc(pi t) / pi + C..........[Ans.]

To prove the result, just take the derivative of csc(pi t) and prove to yourself it is what is in the integrand.

--

k, then we can rewrite the integrand for the second on to make it more inviting to substitution:

(b) csc(pi t) cot(pi t) dt

I think it might fly with your teacher to just do your substitution as u = pi t, du / pi = dt, so you get

(1/pi) csc(u) cot(u) du

and then recognize the derivative.

Additionally you can write these in terms of sines and cosines:

(1/pi) (1/sin(u))(cos(u)/sin(u)) du

= (1/pi) cos(u) / sin^2(u) du

let v = sin(u), dv = cos(u) du, so we have

= (1/pi) dv / v^2

integrating

= -(1/pi) 1 / v + C

= -(1/pi) 1 / sin(u) + C

= -(1/pi) csc(u) + C

= -(1/pi) csc(pi t) + C............[Ans.]

for help visit link:

math.tutorvista.com/

http://en.wikipedia.org/wiki/Learn_To_Be