Exercises

These exercises are exactly the same as those in Section 8.4, but the instructions are different. Use one or more of the following tools to evaluate each of the following integrals (or find antiderivatives in Exercise 4). Be sure to check your answers in the most appropriate way you can think of. Note that the each tool expects a function of \(x\).

1. 
   

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

2. 
   

   \(\displaystyle\int_0^{\,1} x^2 \sqrt{1+x^3}\,dx\)	\(\displaystyle\int_0^{\,1}\, t\, e^{-\frac{t^2}{2}} \,dt\)	\(\displaystyle\int_0^{\,\pi}\,\sin^2 \theta \,d \theta\)
   \(\displaystyle\int_0^{\,1}\,x \sqrt{x^2+2} \,dx\)	\(\displaystyle\int_0^{\,\pi}\,\cos^2 x\, \sin\,x \,dx\)	\(\displaystyle\int_0^{\,1}\,x \sqrt{9-x^2} \,dx\)
   \(\displaystyle\int_0^{\,1}\,\frac{x}{1+x^2} \,dx\)	\(\displaystyle\int_0^{\,1}\,\frac{x}{\sqrt{1+x^2}} \,dx\)	\(\displaystyle\int_{-1}^{\,1}\,\sqrt{1-\frac{x}{4}} \,dx\)
   \(\displaystyle\int_0^{\,1}\,\frac{x}{\sqrt{9-x^2}} \,dx\)	\(\displaystyle\int_0^{\,1}\,\sqrt{9-x^2} \,dx\)	\(\displaystyle\int_0^{\,1}\,\frac{x^2}{\sqrt{9-x^2}} \,dx\)

3. 
   

   a.   \(\displaystyle\int\,t\,\sin\,2t\,dt\)	b.   \(\displaystyle\int\,t^2\,\sin\,2t\,dt\)	c.   \(\displaystyle\int\,t^3\,\sin\,2t\,dt\)
   d.   \(\displaystyle\int\,\ln\,t\,dt\)	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.   \(\displaystyle\frac{\theta}{3}\,\sin\,\theta\)	b.   \(\displaystyle\theta\,\sin\,\frac{\theta}{3}\)	c.   \(\sin^{-1}\,3x\)
   d.   \((t-2) e^{-t}\)	e.   \((t^2-2) e^{-t}\)	f.   \(x\,\sin^{-1}\,3x\)

5. 
   

   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. 
   

   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. 
   

   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.   \(\displaystyle\int_0^{\,1}\,\frac{x}{\left(1+x^2\right)^2}\,dx\)