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Calculus/Evaluating an integral

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Expert: - 8/3/2010


Question

Integral
How would I go about evaluating this integral?

Answer
On some problems like this, use a u substitution with u = √x.
Thus, du = 1/(2√x) dx, and x^2 + 6 = u^4 + 6, but there is an easier way...

Note that √x = x^0.5.

Break this into ∫x^2/x^0.5 dx + ∫6/x^0.5 dx.
This transforms into ∫x^(2 - 0.5) dx + ∫6/x^0.5 dx.
This is the same as ∫x^1.5 dx  + 6∫x^(-0.5) dx.

To integrate, add 1 to the power and divide by the new power.
This will gives us A1 for ∫x^1.5 dx and A2 for 6∫x^(-0.5) dx.
There needs to be a C added on at the end when we are done.

Thus, the new power on the 1st term is 3/2 + 1 = 5/2, so the 1st term of the answer is
[x^(5/2)]/(5/2).  Multiplying by (2/5)/(2/5) gives us A1 = 2[x^2.5]/5.

The power on the 2nd term is -1/2 + 1 = 1/2, so the answer 6[x^(1/2)]/(1/2).
Multiplying by (1/2)/(1/2) gives A2 = 12[x^0.5].

The answer to the whole thing is A1 + A2 + C, where C is an unknown constant.
To get the answer, put into the final equation what A1 and A2 are.

Scotto

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