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Expert: - 1/30/2009


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QUESTION: Find the exact value of the integration of x(sec^2)x with limits 0 to pi/3. Using the substitution x=tan b, find the integration of db/(1+x^2)^2 with limits 0 to 1,leaving your answer in terms of pi. Can you provides me some links where i can learn more on integration by substitution? I completely dont understand this topic on integration by substitution but normal integration is a bit ok for me. Thanks you very much

ANSWER: Find the exact value of the integration of x(sec^2)x with limits 0 to pi/3.

Having read that, I'll ignore the rest of the question after reading it.

This is a u-dv problem.

Let u = x, dv = sec^2(x) dx.
It can be seen that du = dx and v = tan(x).
From here, we know that
⌠      ⌠
⌡u dv = uv -   ⌡ v du.

That is, xsec²(x)(from 0 to pi/3) -
⌠π/3
⌡0  tan(x) dx.

Evaluating the integral gives sec²(x) with the limits.

Putting x sec²(x) with the result of the integral gives
(x-1)sex²(x) (eval 0 to π/3).

Know sec is 1/cos and evaluate away to get the answer.


---------- FOLLOW-UP ----------

QUESTION:  The 1st part i got this (120-(3)^.5)/2. The 2nd part, i cant see some of the symbol as i was using mobile phone. Can you use normal symbol that can be seen in mobile phone? I dont know what to put at db as the question said integrate db/(1+x^2)^2. Thanks

Answer
I'll restate the answer with normal characters.
I() = integral

ANSWER: Find the exact value of the integration
of x(sec^2)x with limits 0 to pi/3.

This is a u-dv problem.

Let u = x, dv = sec^2(x) dx.
It can be seen that du = dx and v = tan(x).
From here, we know that
I(u)dv = uv - I(v)du.

That is, xsec^2(x)(from 0 to pi/3) -
I(tan(x))dx from 0 to pi/3.

Evaluating the integral gives sec^2(x) with the limits.

Putting x sec^2(x) with the result of the integral gives
(x-1)sex^2(x) (eval 0 to pi/3).

I know sec(x) is 1/cos(x) and take the value at pi/3 - value at 0.


When the function is x*sec^2(x), it makes no sense to use u=tan(x)
as a substitution.

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