4.5 Integration by Substitution
√
R
Motivating example: 3x2 x3 + 1dx.
key: Find a term whose derivative also appears in the integrand. Try the inside of a
composite function.
R 0
d
0
0
(f
(g(x)))
=
f
(g(x))g
(x)
⇒
f (g(x))g 0 (x)dx = f (g(x)).
dx
R
R
ex: 2x(x2 + 1)4 dx; sec2 (x)(tan(x) + 3)dx;
Note: it’s also ok if the derivative that appears differs by a constant multiple:
R √
R
ex: x2 x3 + 1dx; cos(5x)dx.
R √
Sometimes a clever change of variables will suffice: x 2x − 1dx.
General power rule for integration:
ex:
R
sin3 (x)cos(x)dx;
R
R
(g(x))n g 0 (x)dx =
g(x)n+1
n+1
+ C.
−4x
dx.
(1−2x2 )2
Change of variables for definite integrals: If u = g(x) has a cts derivative on the closed
Rb
R g(b)
interval [a, b], and f is cts on the range of g, then a f (g(x))g 0 (x)dx = g(a) f (u)du.
ex’s:
R1
0
x(x2 + 1)3 dx;
R5
1
√ x
dx;
2x−1
Thm: Let f be integrable on [−a,
R aa]:
Ra
i.) If f is an even function, then −a f (x)dx = 2 0 f (x)dx.
Ra
ii.) If f is an odd function, then −a f (x)dx = 0.
ex:
R3
x5 − 5x3 dx;
−3
R4
−4
|x|dx.
Random book problems.
1