Ximpledu

Definite Integral

See how to solve a definite integral.
13 examples and their solutions.

Definite Integral: Meaning

Definition


abf(x) dx
A definite integral is the area of the region below y = f(x).
(when y = f(x) is above the x-axis.)
It's the sum of the area of the thinly sliced rectangle.
Riemann Sum

Definite Integral: How to Solve

Formula

abf(x) dx = [F(x)]ab
= F(b) - F(a)
a: Lower limit
b: Upper limit

How to solve:
1. Find F(x): Indefinite integral of f(x)
2. Put b and a into F(x).

The result of a definite integral is a value.
(F(b) - F(a) doesn't have a variable.)

[F(x)]ab = F(b) + C - [F(a) + C]
= F(b) - F(a)
→ You don't have to write +C
when solving a definite integral.

Example

-12x3 dx
Solution

Example

13(6x2 - 2x + 5) dx
Solution

Example

01dx3x + 1
Solution

Definite Integral: Property

Variable in the Definite Integral

abf(x) dx = abf(y) dy
The result of a definite integral is a constant.
So the variable in the definite integral
doesn't affect the definite integral value.
(If the lower/upper limit is a variable,
then it's a different story.)

Upper Limit = Lower Limit

aaf(x) dx = 0

Example

1xf(t) dt = x2 + px
p = ?
Solution

Same Limits

abf(x) dx ± abg(x) dx
= ab(f(x) ± g(x)) dx

Example

13(x2 + x - 1) dx + 132y2 dy - 13z dz
Solution

Connected Limits

acf(x) dx + cbf(x) dx
= abf(x) dx
f(x) should be the same.

Example

01(2x + 1) dx + 14(2x + 1) dx
Solution

Switching Upper Limit and Lower Limit

abf(x) dx = -baf(x) dx

Example

085x4 dx - 285x4 dx
Solution

Definite Integral of an Odd/Even Function

Odd Function


x, x3, x5
sin x, tan x
sinh x
An odd function is a function
whose graph is symmetric to the origin.

Even Function


x2, x4, x6
cos x
cosh x
An even function is a function
whose graph is symmetric to the y-axis.

Property


-aa(Odd function) dx = 0

-aa(Even function) dx = 2⋅0a(Even function) dx

Example

-11(5x4 + 8x3 - 6x2 - 2x + 7) dx
Solution

Example

-π3π3(sin θ + cos θ) dθ
Solution

Integral by Substitution

Example

0π2esin x cos x dx
Solution

Example

ee4dxx ln x
Solution

Integral by Parts

Formula

abuv' dx = [uv]ab - abu'v dx
Integral by Parts: Indefinite Integral

How to Determine u and v'

ln x, loga x
x, x2, x3, ...
sin x, cos x
ex, ax
uv'
Upper function → u
(= The integral is complex.)
Lower function → v'
(= The integral is simple.)

Example

0π3x sin x dx
Solution

Example

1e(ln x)2 dx
Solution