Problems

1. In Activity 5 you explored two different substitutions for transforming the integral

   ∫ cos 2 x dx cos 4 x+ cos 2 x+1 .

   Now try the substitution \(u=\cos^3x\). You should find the resulting integral is simple enough that you can write down the answer by inspection — no CAS required. (Hint: Your replacement for \(dx\) in terms of \(du\) will be easier if you multiply numerator and denominator by \(\sin\,x\).)

2. One of the tools used by a CAS for integration is to identify a "form" for an antiderivative with some undetermined coefficients, and then find the coefficients by differentiating the form and matching terms. In this problem we ask you to carry out this procedure by hand in a relatively simple case.
   1. To find an antiderivative of \(f(x)=x e^x\), we suppose that the form of the answer is \(F(x)+C\), where \(F(x)=Axe^x+Be^x\). Find the constants \(A\) and \(B\) by differentiation.
   2. Repeat the procedure for \(g(x)=2xe^x\).
   3. What modifications are needed in the procedure for it to work with \(h(x)=xe^{2x}\)?
   4. Try to modify the procedure so that it will work for the function \(k\) defined by \(k(x)=x^2e^x\).

   This problem is adapted from Calculus Problems for a New Century, ed. by R. Fraga, MAA Notes No. 28, 1993.

The remaining problems are taken from "Integration on Computer Algebra Systems" by Kevin Charlwood, The Electronic Journal of Mathematics and Technology, Volume 2, Number 3, 2008. For each problem, we provide a link to Wolfram|Alpha and a Sage Indefinite Integral tool. Either may succeed or fail at evaluating each of these integrals. If at least one of them succeeds, you may still need some ingenuity to check the answers. And if the two CAS systems produce different answers, you may need to confirm that they differ by a constant. To that end, we also provide the Graph Difference tool where it might be needed.

3. \(\displaystyle\int \frac{x\,\sin^{-1}x}{\sqrt{1-x^2}} \,dx\). Compare the CAS solution with integration by parts — which you can probably complete by hand.

4. \(\displaystyle\int \frac{x^3\,e^{\arcsin\,x}}{\sqrt{1-x^2}}\,dx\). If your CAS can't evaluate this, try substituting \(x=\sin\,u\). Remember that you're not finished until you make the reverse substitution in the answer, in order to express the answer in terms of \(x\).

5. \(\displaystyle\int \frac{\sqrt{1+x^3}}{x}\,dx\)