5.3. INTEGRATION BY SUBSTITUTION
297
A key part of choosing the expression in x to be represented by u is the identification of
a function-derivative pair. To do so, we often look for an “inner” function g(x) that is
part of a composite function, while investigating whether g ′ (x) (or a constant multiple
of g ′ (x)) is present as a multiplying factor of the integrand.
Exercises
1. This problem centers on finding antiderivatives for the basic trigonometric functions
other than sin(x) and cos(x).
(a) Consider the indefinite integral
tan(x) dx. By rewriting the integrand as
tan(x) =
sin(x)
cos(x) and identifying an appropriate function-derivative pair, make a
u-substitution and hence evaluate
tan(x) dx.
(b) In a similar way, evaluate
cot(x) dx.
(c) Consider the indefinite integral
sec 2 (x) + sec(x) tan(x)
sec(x) + tan(x)
dx.
Evaluate this integral using the substitution u = sec(x) + tan(x).
(d) Simplify the integrand in (c) by factoring the numerator. What is a far simpler
way to write the integrand?
(e) Combine your work in (c) and (d) to determine
sec(x) dx.
(f) Using (c)-(e) as a guide, evaluate
csc(x) dx.
2. Consider the indefinite integral
x
√
x − 1 dx.
(a) At first glance, this integrand may not seem suited to substitution due to the
presence of x in separate locations in the integrand. Nonetheless, using the
composite function
√
x − 1 as a guide, let u = x − 1. Determine expressions for
both x and dx in terms of u.
(b) Convert the given integral in x to a new integral in u.
(c) Evaluate the integral in (b) by noting that
√
u = u 1/2 and observing that it is
now possible to rewrite the integrand in u by expanding through multiplication.
(d) Evaluate each of the integrals
x
2
√
x − 1 dx and
x
√
x 2 − 1 dx. Write a
297
A key part of choosing the expression in x to be represented by u is the identification of
a function-derivative pair. To do so, we often look for an “inner” function g(x) that is
part of a composite function, while investigating whether g ′ (x) (or a constant multiple
of g ′ (x)) is present as a multiplying factor of the integrand.
Exercises
1. This problem centers on finding antiderivatives for the basic trigonometric functions
other than sin(x) and cos(x).
(a) Consider the indefinite integral
tan(x) dx. By rewriting the integrand as
tan(x) =
sin(x)
cos(x) and identifying an appropriate function-derivative pair, make a
u-substitution and hence evaluate
tan(x) dx.
(b) In a similar way, evaluate
cot(x) dx.
(c) Consider the indefinite integral
sec 2 (x) + sec(x) tan(x)
sec(x) + tan(x)
dx.
Evaluate this integral using the substitution u = sec(x) + tan(x).
(d) Simplify the integrand in (c) by factoring the numerator. What is a far simpler
way to write the integrand?
(e) Combine your work in (c) and (d) to determine
sec(x) dx.
(f) Using (c)-(e) as a guide, evaluate
csc(x) dx.
2. Consider the indefinite integral
x
√
x − 1 dx.
(a) At first glance, this integrand may not seem suited to substitution due to the
presence of x in separate locations in the integrand. Nonetheless, using the
composite function
√
x − 1 as a guide, let u = x − 1. Determine expressions for
both x and dx in terms of u.
(b) Convert the given integral in x to a new integral in u.
(c) Evaluate the integral in (b) by noting that
√
u = u 1/2 and observing that it is
now possible to rewrite the integrand in u by expanding through multiplication.
(d) Evaluate each of the integrals
x
2
√
x − 1 dx and
x
√
x 2 − 1 dx. Write a