Introduction to Integration
Definition: Integration is the process of finding the integral of a function, which
can be interpreted as the area under the curve of the function on a given
interval. The integral of a function f(x)f(x)f(x) over an interval [a,b][a, b][a,b] is
denoted as ∫abf(x) dx\int_a^b f(x) \, dx∫abf(x)dx.
Conceptual Understanding:
Antiderivative: The integral of a function f(x)f(x)f(x) is also known as the
antiderivative of f(x)f(x)f(x), which is a function F(x)F(x)F(x) such that F′
(x)=f(x)F'(x) = f(x)F′(x)=f(x).
Area Interpretation: The definite integral ∫abf(x) dx\int_a^b f(x) \, dx∫ab
f(x)dx represents the signed area between the function f(x)f(x)f(x) and the
x-axis from x=ax = ax=a to x=bx = bx=b.
2. Fundamental Theorem of Calculus
1. First Part of the Fundamental Theorem: If F(x)F(x)F(x) is an antiderivative
of f(x)f(x)f(x) on [a,b][a, b][a,b], then: ∫abf(x) dx=F(b)−F(a)\int_a^b f(x) \, dx =
F(b) - F(a)∫abf(x)dx=F(b)−F(a)
2. Second Part of the Fundamental Theorem: If f(x)f(x)f(x) is continuous on
[a,b][a, b][a,b], then the function F(x)F(x)F(x) defined by: F(x)=∫axf(t) dtF(x) = \
int_a^x f(t) \, dtF(x)=∫axf(t)dt is continuous on [a,b][a, b][a,b], differentiable on
(a,b)(a, b)(a,b), and F′(x)=f(x)F'(x) = f(x)F′(x)=f(x).
3. Basic Rules of Integration
1. Power Rule: If f(x)=xnf(x) = x^nf(x)=xn where n≠−1n \neq -1n =−1, then:
∫xn dx=xn+1n+1+C\int x^n \, dx = \frac{x^{n+1}}{n+1} + C∫xndx=n+1xn+1
+C where CCC is the constant of integration.
2. Constant Multiple Rule: If f(x)=c⋅g(x)f(x) = c \cdot g(x)f(x)=c⋅g(x), then:
∫c⋅g(x) dx=c⋅∫g(x) dx\int c \cdot g(x) \, dx = c \cdot \int g(x) \,
dx∫c⋅g(x)dx=c⋅∫g(x)dx
3. Sum Rule: If f(x)=g(x)+h(x)f(x) = g(x) + h(x)f(x)=g(x)+h(x), then: ∫[g(x)
+h(x)] dx=∫g(x) dx+∫h(x) dx\int [g(x) + h(x)] \, dx = \int g(x) \, dx + \int h(x) \,
dx∫[g(x)+h(x)]dx=∫g(x)dx+∫h(x)dx
4. Difference Rule: If f(x)=g(x)−h(x)f(x) = g(x) - h(x)f(x)=g(x)−h(x), then:
∫[g(x)−h(x)] dx=∫g(x) dx−∫h(x) dx\int [g(x) - h(x)] \, dx = \int g(x) \, dx - \int
h(x) \, dx∫[g(x)−h(x)]dx=∫g(x)dx−∫h(x)dx
4. Integration Techniques
1. Substitution: Used to simplify integrals by substituting a new variable.
Process: Let u=g(x)u = g(x)u=g(x). Then du=g′(x) dxdu = g'(x) \, dxdu=g
′(x)dx, and the integral becomes: ∫f(g(x))⋅g′(x) dx=∫f(u) du\int f(g(x)) \cdot
g'(x) \, dx = \int f(u) \, du∫f(g(x))⋅g′(x)dx=∫f(u)du
2. Integration by Parts: Based on the product rule for differentiation.
Formula: ∫u dv=uv−∫v du\int u \, dv = uv - \int v \, du∫udv=uv−∫vdu
where uuu and vvv are functions of xxx, and dududu and dvdvdv are their
respective differentials.
3. Partial Fraction Decomposition: Used for rational functions.
Process: Express the rational function as a sum of simpler fractions. Then
integrate each fraction separately.
Example: For P(x)Q(x)\frac{P(x)}{Q(x)}Q(x)P(x), where Q(x)Q(x)Q(x) can
be factored into simpler linear or quadratic terms.
4. Trigonometric Substitution: Used for integrals involving square roots of
quadratic expressions.
Common Substitutions:
o For a2−x2\sqrt{a^2 - x^2}a2−x2, use x=asin(θ)x = a \sin(\
theta)x=asin(θ)
o For a2+x2\sqrt{a^2 + x^2}a2+x2, use x=atan(θ)x = a \tan(\
theta)x=atan(θ)
o For x2−a2\sqrt{x^2 - a^2}x2−a2, use x=asec(θ)x = a \sec(\
theta)x=asec(θ)
5. Integration by Trigonometric Identities: Use identities to simplify
integrals involving trigonometric functions.
Example: For ∫sin2(x) dx\int \sin^2(x) \, dx∫sin2(x)dx, use the identity
sin2(x)=1−cos(2x)2\sin^2(x) = \frac{1 - \cos(2x)}{2}sin2(x)=21−cos(2x).
6. Numerical Integration: Used when an integral cannot be expressed in
closed form.
Methods: Trapezoidal rule, Simpson’s rule.
5. Special Integrals
1. Exponential Functions:
∫ex dx=ex+C\int e^x \, dx = e^x + C∫exdx=ex+C
∫ax dx=axln(a)+C\int a^x \, dx = \frac{a^x}{\ln(a)} + C∫axdx=ln(a)ax
+C, where aaa is a positive constant.
2. Logarithmic Functions:
∫ln(x) dx=xln(x)−x+C\int \ln(x) \, dx = x \ln(x) - x + C∫ln(x)dx=xln(x)−x+C
∫1x dx=ln∣x∣+C\int \frac{1}{x} \, dx = \ln|x| + C∫x1dx=ln∣x∣+C
3. Trigonometric Functions:
∫sin(x) dx=−cos(x)+C\int \sin(x) \, dx = -\cos(x) + C∫sin(x)dx=−cos(x)+C
∫cos(x) dx=sin(x)+C\int \cos(x) \, dx = \sin(x) + C∫cos(x)dx=sin(x)+C
∫tan(x) dx=ln∣sec(x)∣+C\int \tan(x) \, dx = \ln|\sec(x)| +
C∫tan(x)dx=ln∣sec(x)∣+C
∫cot(x) dx=ln∣sin(x)∣+C\int \cot(x) \, dx = \ln|\sin(x)| +
C∫cot(x)dx=ln∣sin(x)∣+C
∫sec(x) dx=ln∣sec(x)+tan(x)∣+C\int \sec(x) \, dx = \ln|\sec(x) + \tan(x)| +
C∫sec(x)dx=ln∣sec(x)+tan(x)∣+C
∫csc(x) dx=ln∣csc(x)−cot(x)∣+C\int \csc(x) \, dx = \ln|\csc(x) - \cot(x)| +
C∫csc(x)dx=ln∣csc(x)−cot(x)∣+C
4. Inverse Trigonometric Functions:
∫11−x2 dx=arcsin(x)+C\int \frac{1}{\sqrt{1 - x^2}} \, dx = \arcsin(x) +
C∫1−x21dx=arcsin(x)+C
∫11+x2 dx=arctan(x)+C\int \frac{1}{1 + x^2} \, dx = \arctan(x) +
C∫1+x21dx=arctan(x)+C
∫1∣x∣x2−1 dx=\arcsec(∣x∣)+C\int \frac{1}{|x| \sqrt{x^2 - 1}} \, dx = \
arcsec(|x|) + C∫∣x∣x2−11dx=\arcsec(∣x∣)+C
6. Improper Integrals
Definition: Improper integrals are those where either the interval of integration
is infinite or the integrand has an infinite discontinuity within the interval.
1. Infinite Limits:
Example: ∫a∞f(x) dx\int_a^\infty f(x) \, dx∫a∞f(x)dx Evaluate the limit as
b→∞b \to \inftyb→∞: ∫a∞f(x) dx=limb→∞∫abf(x) dx\int_a^\infty f(x) \, dx = \
lim_{b \to \infty} \int_a^b f(x) \, dx∫a∞f(x)dx=limb→∞∫abf(x)dx
2. Discontinuous Integrands:
Example: ∫ab1x−a dx\int_a^b \frac{1}{\sqrt{x - a}} \, dx∫abx−a1dx
Evaluate the limit as xxx approaches the point of discontinuity: ∫ab1x−a
dx=limϵ→0+[∫aa+ϵ1x−a dx+∫a+ϵb1x−a dx]\int_a^b \frac{1}{\sqrt{x -
a}} \, dx = \lim_{\epsilon \to 0^+} \left[ \int_a^{a+\epsilon} \frac{1}{\
sqrt{x - a}} \, dx + \int_{a+\epsilon}^b \frac{1}{\sqrt{x - a}} \, dx \
right]∫abx−a1dx=limϵ→0+[∫aa+ϵx−a1dx+∫a+ϵbx−a1dx]
7. Applications of Integration
1. Area Between Curves: To find the area between two curves f(x)f(x)f(x) and
g(x)g(x)g(x), where f(x)≥g(x)f(x) \geq g(x)f(x)≥g(x): Area=∫ab[f(x)−g(x)] dx\
text{Area} = \int_a^b [f(x) - g(x)] \, dxArea=∫ab[f(x)−g(x)]dx
2. Volume of Solids of Revolution: Using the disk method: V=π∫ab[f(x)]2 dxV
= \pi \int_a^b [f(x)]^2 \, dxV=π∫ab[f(x)]2dx Using the washer method:
V=π∫ab[[R(x)]2−[r(x)]2] dxV = \pi \int_a^b \left[ [R(x)]^2 - [r(x)]^2 \right] \,
dxV=π∫ab[[R(x)]2−[r(x)]2]dx where R(x)R(x)R(x) and r(x)r(x)r(x) are the outer
and inner radii.
3. Arc Length: To find the arc length of a curve y=f(x)y = f(x)y=f(x) from x=ax
= ax=a to x=bx = bx=b: Arc Length=∫ab1+[f′(x)]2 dx\text{Arc Length} = \
int_a^b \sqrt{1 + [f'(x)]^2} \, dxArc Length=∫ab1+[f′(x)]2dx
4. Surface Area of Revolution: For revolving y=f(x)y = f(x)y=f(x) around the
x-axis: Surface Area=2π∫abf(x)1+[f′(x)]2 dx\text{Surface Area} = 2\pi \int_a^b
f(x) \sqrt{1 + [f'(x)]^2} \, dxSurface Area=2π∫abf(x)1+[f′(x)]2dx
