Substitutions in Integrals

Question 1

The first Chebyshev substitution

Answer 1

∫x^p * (a + b*x^q)^r dx where r is an integer -> substitution x = t^n , n is the common denominator of p and q.

Question 2

The second Chebyshev substitution

Answer 2

∫x^p * (a + bx^q)^r dx where (p + 1) / q is an integer -> substitution (a + bx^q) = t^n , n is the denominator of r

Question 3

The third Chebyshev substitution

Answer 3

∫x^p * (a + b*x^q)^r dx where (p + 1) / q + r is an integer -> substitution (1/x^q + b) = t^n , n is the denominator of r.

Question 4

Universal trig substitution

Answer 4

tan x/2 = t , -π + 2πk < x < π + 2πk, (k is an integer) 
Then 
x = 2arctan t, 
dx = 2dt / (1 + t^2), 
sin x = 2t / (1 + t^2),
cos x = (1 - t^2) / (1 + t^2)

Question 5

R(-sinx, cosx) = -R(sinx, cosx)

Answer 5

R(-sinx, cosx) = -R(sinx, cosx) then we substitute cosx = t

Question 6

R(sinx, -cosx) = -R(sinx, cosx)

Answer 6

R(-sinx, cosx) = -R(sinx, cosx) then we substitute sinx = t

Question 7

R(-sinx, -cosx) = R(sinx, cosx)

Answer 7

R(-sinx, -cosx) = R(sinx, cosx) then we substitute tanx = t or atanx = t.