<p>What's the antiderivative of x^3(1-x^2)^(1/2)? I tried integration by parts w/ the tabular method but that is just messy...</p>
<p>And another non-antiderivative question: W/ an interest rate of 5% annually, compounded continuously, at what CONSTANT RATE of dollars/year should you invest money so that you end w/ $20,000 after 20 years. I was fooled initially and used the formula for natural growth/decay and instead solved for a sum of money that you should invest, not a rate of investment. </p>
<p>I would integrate by parts - probably the simplest would be to say
u = (1-x^2)^1/2
and
u^2 = (1-x^2)</p>
<p>x^2 = 1-u^2
and separate it to:
f(x) = x^2 * x * (1-x^2)^(1/2) then</p>
<p>du = 1/2 * (1-x^2)^1/2 * -2x dx</p>
<p>simplify to</p>
<p>du = 1/2 U * -2x dx
-du/U = xdx</p>
<p>then you should replace all the x's with U's be home free for an easy integration with integral of -(1-u^2)*u^2 du I believe. Again quickly done so check the work.</p>
<p>Thanks, ABlestMom, but I'm still not understanding the integration by parts-- shouldn't du = 1/2 * (1-x^2)^(-1/2) * -2x dx? Also in the 2nd problem there is no initial investment-- the initial investment = the rate of investment for 20 years; you need to figure out the rate of investment per year (ie, 500 dollars/year)</p>