Summary Algebra and Integration in Calculus

Algebra and Integration in Calculus
Algebraic functions and integration are key concepts in calculus. Integration is the process of
finding the antiderivative or the area under a curve. It is the reverse process of
differentiation and has numerous applications in science, engineering, and economics.

Key Concepts in Algebraic Integration
1. Indefinite Integral (Antiderivative):
o The indefinite integral of a function is the general form of its antiderivative. It is
denoted as:∫f(x) dx∫f(x)dx Where f(x)f(x) is the function being integrated, and
indicates the variable of integration.
o Constant of Integration: When integrating, always add a constant CC because
an indefinite integral represents a family of functions.
2. ∫f(x) dx=F(x)+C∫f(x)dx=F(x)+C
Example:
∫xn dx=xn+1n+1+C,(where n≠−1)∫xndx=n+1xn+1+C,(where n=−1)
3. Definite Integral:
o A definite integral represents the area under a curve between two limits, aa and
bb, and is written as:∫abf(x) dx∫abf(x)dx
o It calculates the total accumulated area from x=ax=a to x=bx=b. The result of a
definite integral is a numerical value.
4. Example:
∫13(2x+1) dx∫13(2x+1)dx
Step-by-step:
o Find the antiderivative:∫(2x+1) dx=x2+x+C∫(2x+1)dx=x2+x+C
o Apply limits a=1a=1 and
b=3b=3:[x2+x]13=(32+3)−(12+1)=(9+3)−(1+1)=12−2=10[x2+x]13
=(32+3)−(12+1)=(9+3)−(1+1)=12−2=10
o The area under the curve is 10.
5. Properties of Integrals:
o Linearity: Integration is linear,
meaning:∫[af(x)+bg(x)] dx=a∫f(x) dx+b∫g(x) dx∫[af(x)+bg(x)]dx=a∫f(x)dx+b∫g(x)dx
o Additivity: If the interval of integration is split into smaller subintervals,
then:∫abf(x) dx=∫acf(x) dx+∫cbf(x) dx∫abf(x)dx=∫acf(x)dx+∫cbf(x)dx



Common Algebraic Functions and Their Integrals:
1. Power Rule:
∫xn dx=xn+1n+1+C,n≠−1∫xndx=n+1xn+1+C,n=−1
Example:
∫x3 dx=x44+C∫x3dx=4x4+C
2. Exponential Functions:
∫ex dx=ex+C∫exdx=ex+C
Example:
∫e2x dx=e2x2+C∫e2xdx=2e2x+C

, 3. Logarithmic Functions:
∫1x dx=ln⁡∣x∣+C∫x1dx=ln∣x∣+C
Example:
∫1x+1 dx=ln⁡∣x+1∣+C∫x+11dx=ln∣x+1∣+C
4. Trigonometric Functions:
o ∫sin⁡(x) dx=−cos⁡(x)+C∫sin(x)dx=−cos(x)+C
o ∫cos⁡(x) dx=sin⁡(x)+C∫cos(x)dx=sin(x)+C
o ∫sec⁡2(x) dx=tan⁡(x)+C∫sec2(x)dx=tan(x)+C
5. Inverse Trigonometric Functions:
o ∫11−x2 dx=arcsin⁡(x)+C∫1−x21dx=arcsin(x)+C
o ∫11+x2 dx=arctan⁡(x)+C∫1+x21dx=arctan(x)+C




Integration Techniques:
1. Substitution Rule:
o Use when the integrand is a composite function. If u=g(x)u=g(x),
then:∫f(g(x))⋅g′(x) dx=∫f(u) du∫f(g(x))⋅g′(x)dx=∫f(u)du
2. Example:
∫2x⋅ex2 dx(let u=x2,du=2x dx)∫2x⋅ex2dx(let
u=x2,du=2xdx)∫eu du=eu+C=ex2+C∫eudu=eu+C=ex2+C
3. Integration by Parts:
o This is based on the product rule for differentiation and is useful when integrating
a product of two functions.∫u dv=uv−∫v du∫udv=uv−∫vdu
4. Example:
∫xcos⁡(x) dx∫xcos(x)dx
Let u=xu=x, and dv=cos⁡(x) dxdv=cos(x)dx. Then du=dxdu=dx, and v=sin⁡(x)v=sin(x).
Applying the integration by parts formula:
∫xcos⁡(x) dx=xsin⁡(x)−∫sin⁡(x) dx=xsin⁡(x)+cos⁡(x)+C∫xcos(x)dx=xsin(x)−∫sin(x)dx=xsi
n(x)+cos(x)+C
5. Partial Fraction Decomposition:
o Useful for rational functions where the degree of the numerator is less than the
degree of the denominator. Express the rational function as a sum of simpler
fractions. Example:
6. ∫1x2−1 dx=∫(12(x−1)−12(x+1)) dx∫x2−11dx=∫(2(x−1)1−2(x+1)1)dx
Then integrate each term:
12ln⁡∣x−1∣−12ln⁡∣x+1∣+C21ln∣x−1∣−21ln∣x+1∣+C
7. Trigonometric Substitution:
o Used when the integrand contains square roots of quadratic expressions. Use a
trigonometric identity to simplify the integrand. Example:
8. ∫11−x2 dx∫1−x21dx
Let x=sin⁡(θ)x=sin(θ), so dx=cos⁡(θ) dθdx=cos(θ)dθ. The integral becomes:
∫1cos⁡(θ)⋅cos⁡(θ) dθ=∫dθ=θ+C=arcsin⁡(x)+C∫cos(θ)1⋅cos(θ)dθ=∫dθ=θ+C=arcsin(x)+C



Applications of Integration:
1. Finding Area Under a Curve: