Exercises

1. Evaluate each of the following indefinite integrals. Check your answers by differentiation.
   

   \(\displaystyle\int x^2 \sqrt{1+x^3}\;dx\)	\(\displaystyle\int\; t\,e^{-\frac{t^2}{2}} \,dt\)	\(\displaystyle\int\,\sin^2 \theta\,d \theta\)
   \(\displaystyle\int\,x \sqrt{x^2+2} \;dx\)	\(\displaystyle\int\,\cos^2 x\,\sin\,x \,dx\)	\(\displaystyle\int\,x \sqrt{9-x^2} \;dx\)
   \(\displaystyle\int\,\frac{x}{1+x^2}\,dx\)	\(\displaystyle\int\;\frac{x}{\sqrt{1+x^2}} \;dx\)	\(\displaystyle\int\;\sqrt{1-\frac{x}{4}} \;dx\)
   \(\displaystyle\int\; \frac{x}{\sqrt{9-x^2}} \;dx\)	\(\displaystyle\int\,\sqrt{9-x^2} \;dx\)	\(\displaystyle\int\;\frac{x^2}{\sqrt{9-x^2}} \;dx\)

2. Evaluate exactly each of the following integrals. If you already have an antiderivative from Exercise 1, use it. If not, use the substitution technique for definite integrals. Use the Definite Integral tool to check your work. (Note that the tool expects a function of \(x\).)

   \(\displaystyle\int_0^1 x^2 \sqrt{1+x^3} \;dx\)	\(\displaystyle\int_0^1\, t\,e^{-\frac{t^2}{2}} \,dt\)	\(\displaystyle\int_0^{\pi}\,\sin^2 \theta\,d \theta\)
   \(\displaystyle\int_0^1\,x \sqrt{x^2+2} \;dx\)	\(\displaystyle\int_0^{\pi}\,\cos^2 x\,\sin\,x \,dx\)	\(\displaystyle\int_0^1\,x \sqrt{9-x^2} \;dx\)
   \(\displaystyle\int_0^1\,\frac{x}{1+x^2} \,dx\)	\(\displaystyle\int_0^1\,\frac{x}{\sqrt{1+x^2}} \;dx\)	\(\displaystyle\int_{-1}^1\,\sqrt{1-\frac{x}{4}} \;dx\)
   \(\displaystyle\int_0^1 \,\frac{x}{\sqrt{9-x^2}} \;dx\)	\(\displaystyle\int_0^1\,\sqrt{9-x^2} \;dx\)	\(\displaystyle\int_0^1\,\frac{x^2}{\sqrt{9-x^2}} \;dx\)
3. Use integration by parts to evaluate each of the following indefinite integrals. In each case, think about what gets simpler when you differentiate one of the parts.

   a.   \(\displaystyle\int\,t\,\sin\,2t\,dt\)	b.   \(\displaystyle\int\,t^2\,\sin\,2t\, dt\)	c.   \(\displaystyle\int\,t^3\,\sin\,2t \,dt\)
   d.   \(\displaystyle\int\,\ln\,t \,dt\)   (Hint: Consider \(\ln\,t\) as a product of \(1\) and \(\ln\,t\).)
   e.   \(\displaystyle\int\,t\,\ln\,t\,dt\)	f.   \(\displaystyle\int\,\sin^{-1}\, t\;dt\)	g.   \(\displaystyle\int\,t\,\sin^{-1}\, t\;dt\)

4. Find an antiderivative for each of the following functions.

   a.   \(\frac{\theta}{3}\,\sin\,\theta\)	b.   \(\theta\,\sin\,\frac{\theta}{3}\)	c.   \(\sin^{-1}\,3x\)
   d.   \((t-2) e^{-t}\)	e.   \(\left(t^2-2\right) e^{-t}\)	f.   \(x\,\sin^{-1}\,3x\)

5. Evaluate each of the following integrals. The first step in each case is to think about an appropriate strategy, and then carry it out. If your chosen strategy does not work, the next step is to abandon it and try a different strategy. Check your result by one of the following checking techniques:
   • Differentiate the resulting function.
   • Evaluate a definite integral using the antiderivative, and check with the Definite Integral tool.

   a.   \(\displaystyle\int\, \frac{\cos\,t}{2+\sin\,t} dt\)	b.   \(\displaystyle\int\,\theta^2\,\cos\, 2 \theta\,d \theta\)	c.   \(\displaystyle\int\,\tan^{-1} 2x\,dx\)
   d.   \(\displaystyle\int\,x^2\,\tan^{-1}\, 2x\;dx\)	e.   \(\displaystyle\int\,x \sqrt{x+7}\;dx\)	f.   \(\displaystyle\int\,x \sqrt{x^2+7}\; dx\)
   g.   \(\displaystyle\int\,\frac{\ln\,x}{x} \,dx\)	h.   \(\displaystyle\int\,u\,e^u\,du\)	i.   \(\displaystyle\int\,u\,e^{u^2}\,du\)
6. Evaluate each of the following definite integrals. If you did the corresponding parts of Exercise 5, use the result here. Check your answer with the Definite Integral tool.
   

   a.   \(\displaystyle\int_0^{2 \pi} \frac{\cos\,t}{2+\sin\,t} dt\)	b.   \(\displaystyle\int_0^{\,\pi}\,\theta^2 \,\cos\,2 \theta\,d \theta\)	c.   \(\displaystyle\int_0^{1/2}\tan^{-1} 2x\,dx\)
   d.   \(\displaystyle\int_0^{1/2}x^2\, \tan^{-1}\, 2x\;dx\)	e.   \(\displaystyle\int_0^1 x \sqrt{x+7}\; dx\)	f.   \(\displaystyle\int_0^1 x \sqrt{x^2+7} \;dx\)
   g.   \(\displaystyle\int_1^2\, \frac{\ln\,x}{x}\,dx\)	h.   \(\displaystyle\int_0^1 u\,e^u\,du\)	i.   \(\displaystyle\int_0^1 u\,e^{u^2} \,du\)
7. Evaluate each of the following integrals.

   a.   \(\displaystyle\int_0^1 x^2 \sqrt{4x^3+2}\,dx\)	b.   \(\displaystyle\int\, \left(t^2+\sin\,3t\right)\,dt\)	c.   \(\displaystyle\int\, \frac{7}{\sqrt{s+1}}\,ds\)
   d.   \(\displaystyle\int\,\frac{1}{x^2+4} \,dx\)	e.   \(\displaystyle\int_3^7\,\frac{4}{2y+1} \,dy\)	f.   \(\displaystyle\int_{-\pi/4}^{\,\pi/4} \,\cos\,2x\,dx\)
   g.   \(\displaystyle\int_0^2\,e^{-0.3t}\, dt\)	h.  \(\displaystyle\int\,2x \left(1+x^2\right)^{3/4}dx\)	i.   \(\displaystyle\int\,\sqrt{1-4w^2}\; dw\)
   j.   \(\displaystyle\int_0^{\,\pi/4}\, \sin^2\,6 \theta\,d \theta\)	k.   \(\displaystyle\int_{-\pi}^{\,\pi} \,\cos^2\,3t\,dt\)	l.   \(\displaystyle\int\,t\,e^{5.3t}\,dt\)
   m.   \(\displaystyle\int_0^1\,t\,e^{5.3t} \,dt\)	n.   \(\displaystyle\int\,(t+3)\,e^{5.3t} \,dt\)	o.   \(\displaystyle\int\,t\,\cos\,2t\,dt\)
   p.   \(\displaystyle\int\, t^2\,\sin\,2t\,dt\)	q.   \(\displaystyle\int_0^{\,\pi/2}\,t^2\, \sin\,2t\,dt\)	r.   \(\displaystyle\int_0^1 \,\frac{x}{\left(1+x^2\right)^2}\,dx\)