MATH 2260 Worksheet
Han-Bom Moon
Worksheet Week 1
Review of Chapter 5, from Definition of integral to Substitution method
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1. Evaluate the definite integral
Z
1
3
4x2 + 3
dx.
x2
(a) Simplify the integrand if you can (for instance, use law of exponents).
3
Z
1
4x2 + 3
dx =
x2
Z
3
4 + 3x−2 dx
1
Sometimes such algebraic manipulation is very important to computation.
(b) Find the antiderivative and use the fundamental theorem of calculus.
Z 3
3
4 + 3x−2 dx = 4x − 3x−1 1
1
(c) Compute the answer.
4x − 3x−1
3
1
1
= (4 · 3 − 3 · ) − (4 · 1 − 3 · 1) = 10
3
1
MATH 2260 Worksheet
Han-Bom Moon
2. Evaluate the definite integral
Z
−1
−3
Z
−1
−3
=
x5 − 2x
dx.
x3
x5 − 2x
dx =
x3
1 3
x + 2x−1
3
−1
−3
Z
−1
x2 − 2x−2 dx
−3
1
1
2
= (− − 2) − ( · (−27) +
)
3
3
−3
=
22
3
3. Evaluate the definite integral
Z
π
(1 + cos x)dx.
0
Z
0
π
(1 + cos x)dx = [x + sin x]π0 = (π + sin π) − (0 + sin 0) = π
2
MATH 2260 Worksheet
Han-Bom Moon
4. (Substitution method for indefinite integral) Evaluate the indefinite integral
Z
x(x2 + 3)2 dx.
(a) Define an appropriate variable u in terms of x.
u = x2 + 3
(b) Compute du in terms of x. Note that the symbol du is
du
dx.
dx
du
= 2x ⇒ du = 2xdx
dx
(c) Substitute everything on the integral using u and du.
Z
Z
Z
1
x(x2 + 3)2 dx = (x2 + 3)2 xdx = u2 du
2
After this step, there must be no x in your integral.
(d) Compute given indefinite integral.
Z
1
u3
u2 du =
+C
2
6
(e) Replace u by a function of x you’ve chosen in (a).
(x2 + 3)3
u3
+C =
+C
6
6
Unfortunately, there is no golden rule for finding u, but here is a tip for you: In
step (c), you have to change everything in terms of u. So you must find u such
du
that this substitution is possible. In particular,
or its constant multiple (not
dx
2 3
du
du
,
, · · · ) must be in the integrand. In other words, find u such that
dx
dx
du
you can express given integrand as “something about u times
”. One another
dx
suggestion is that define u as the inside of complicate part, for instance inside of
parentheses, an exponential function or a square root.
3
MATH 2260 Worksheet
Han-Bom Moon
5. Evaluate the indefinite integral, following similar steps in previous problem.
Z
x5 (x6 − 2)25 dx
u = x6 − 2
du
= 6x5 ⇒ du = 6x5 dx
dx
Z
Z
Z
1
5 6
25
6
25 5
x (x − 2) dx = (x − 2) x dx = u25 du
6
=
u26
u26
(x6 − 2)26
+C =
+C =
+C
6 · 26
156
156
6. Evaluate the indefinite integral, following similar steps in previous problem.
Z
3
x2 ex dx
u = x3
du
= 3x2 ⇒ du = 3x2 dx
dx
Z
Z
Z
1
2 x3
x3 2
x e dx = e x dx = eu du
3
1
1 3
= eu + C = ex + C
3
3
4
MATH 2260 Worksheet
Han-Bom Moon
7. (Substitution method for definite integral) Evaluate the definite integral
Z
e7
1
dx
.
x(1 + ln x)
(a) Define appropriate variable u in terms of x.
u = 1 + ln x
(b) Compute du in terms of x.
du
1
1
= ⇒ du = dx
dx
x
x
(c) Find two endpoints u(1) and u(e7 ).
u(e7 ) = 1 + ln e7 = 1 + 7 = 8
u(1) = 1 + ln 1 = 1,
(d) Substitute everything (including endpoints) on the integral using u and du.
Z
1
e7
dx
=
x(1 + ln x)
Z
1
e7
1
dx =
(1 + ln x)x
Z
8
1
1
du
u
Again, when you apply the substitution method, you must replace everything about x by something about u.
(e) Compute the definite integral.
Z
1
8
1
du = [ln |u|]81 = ln 8 − ln 1 = ln 8
u
5
MATH 2260 Worksheet
Han-Bom Moon
8. Evaluate the definite integral
1
Z
2x
dx
(4 + x2 )2
0
following similar steps in the previous problem.
u = 4 + x2
du
= 2x ⇒ du = 2xdx
dx
u(0) = 4,
Z
0
1
u(1) = 4 + 1 = 5
2x
dx =
(4 + x2 )2
Z
5
4
1
du =
u2
5
Z
u−2 du
4
5
1 1
1
= −u−1 4 = −5−1 − (−4−1 ) = − + =
5 4
20
9. Evaluate the definite integral
Z
1
x
p
1 − x2 dx
0
following similar steps in the previous problem.
u = 1 − x2
du
1
= −2x ⇒ du = −2xdx, − du = xdx
dx
2
u(1) = 1 − 1 = 0
u(0) = 1,
Z
0
1
Z
p
2
x 1 − x dx =
0√
1
1
u(− )du =
2
1 3
=
u2
3
1
=
0
6
Z
1
√
0
1
1
−0=
3
3
u
du =
2
Z
0
1
1
u2
du
2