James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

514 Chapter 7 Techniques of Integration
(b) Based on the pattern of your responses in part (a), guess the value of the integral
y
1
sx 1 adsx 1 bd dx
if a ± b. What if a − b?
(c) Check your guess by asking your CAS to evaluate the integral in part (b). Then prove it
using partial fractions.
2. (a) Use a computer algebra system to evaluate the following integrals.
(i) y sin x cos 2x dx (ii) y sin 3x cos 7x dx (iii) y sin 8x cos 3x dx
(b) Based on the pattern of your responses in part (a), guess the value of the integral
y sin ax cos bx dx
(c) Check your guess with a CAS. Then prove it using the techniques of Section 7.2.
For what values of a and b is it valid?
3. (a) Use a computer algebra system to evaluate the following integrals.
(i) y ln x dx (ii) y x ln x dx (iii) y x 2 ln x dx
(iv) y x 3 ln x dx
(v) y x 7 ln x dx
(b) Based on the pattern of your responses in part (a), guess the value of
y x n ln x dx
(c) Use integration by parts to prove the conjecture that you made in part (b). For what values
of n is it valid?
4. (a) Use a computer algebra system to evaluate the following integrals.
(i) y xe x dx (ii) y x 2 e x dx (iii) y x 3 e x dx
(iv) y x 4 e x dx
(v) y x 5 e x dx
(b) Based on the pattern of your responses in part (a), guess the value of y x 6 e x dx. Then use
your CAS to check your guess.
(c) Based on the patterns in parts (a) and (b), make a conjecture as to the value of the
integral
y x n e x dx
when n is a positive integer.
(d) Use mathematical induction to prove the conjecture you made in part (c).
There are two situations in which it is impossible to find the exact value of a definite
integral.
The first situation arises from the fact that in order to evaluate y b a
f sxd dx using the
Fun damental Theorem of Calculus we need to know an antiderivative of f. Sometimes,
however, it is difficult, or even impossible, to find an antiderivative (see Section 7.5). For
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