Date: Apr 26, 2013 1:44 PM
Author: Axel Vogt
Subject: integration test suite / Chap 7

These are the excercises for Chap 7 in Tomfeev's book: p.334, # 1 - # 4,
p.342/343 #5 - # 9, p.344, # 10, # 11).

Excercise 3 has a typo, the solution should not start with 5/48*x^6 (correction
found with Maple)

In excercise 8 and 9 the author gives an integral, which Maple knows in terms
of polylogarithm (a matter of taste how to write it).

Maple finds all the solution (have not checked for compact results).

L:= [
#1
Int(x^2*cos(x)^5,x) =
1/200*x*cos(5*x) + (1/80*x^2-1/1000)*sin(5*x) +5/72*x*cos(3*x) +
(5/48*x^2-5/216)*sin(3*x) + 5/4*x*cos(x)+(5/8*x^2-5/4)*sin(x),
#2
Int(x^3*sin(x)^3,x) =
1/12*(x^3-2/3*x)*cos(3*x) - 1/12*(x^2-2/9)*sin(3*x) -
3/4*(x^3-6*x)*cos(x) + 9/4*(x^2-2)*sin(x),
#3
Int(x^2*sin(x)^6,x) = 5/48*x^3 - # corrected version, 5/48*x^6 ... is a typo
1/192*(x^2-1/18)*sin(6*x) - 1/576*x*cos(6*x) +
3/64*(x^2-1/8)*sin(4*x) + 3/128*x*cos(4*x) -
15/64*(x^2-1/2)*sin(2*x) - 15/64*x*cos(2*x),
#4
Int(x^2*sin(x)^2*cos(x),x) =
1/3*x^2*sin(x)^3 - 1/18*x*cos(3*x) +
1/54*sin(3*x)+1/2*x*cos(x)-1/2*sin(x),

#5
Int(x*cos(x)^4/sin(x)^2,x) =
-x*cos(x)*(1/2*sin(x)+1/sin(x)) +1/4*sin(x)^2 + ln(sin(x)) - 3/4*x^2,
#6
Int(x*sin(x)^3/cos(x)^4,x) =
x*(1/3/cos(x)^3 - 1/cos(x)) - 1/6*sin(x)/cos(x)^2 + 5/6*ln(tan(Pi/4+x/2)),

#7
Int(x*sin(x)/cos(x)^3,x) =
x/2/cos(x)^2 - 1/2*tan(x),
#8
Int(x*sin(x)^3/cos(x),x) =
1/4*x*cos(2*x) - 1/8*sin(2*x) + Int(x*tan(x), x),
#9
Int(x*sin(x)^3/cos(x)^3, x) =
x/2/cos(x)^2 - 1/2*tan(x) - Int(x*tan(x), x),

#10
Int((2*x+sin(2*x))/(x*sin(x)+cos(x))^2,x) =
-2*cos(x)/(x*sin(x)+cos(x)),
#11
Int((x/(x*cos(x)-sin(x)))^2,x) =
(x*sin(x)+cos(x))/(x*cos(x)-sin(x))
]: