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Integral Calculus

Basic Mathematics

Informatics Engineering

Gunadarma University

Overview of Integral Calculus Integral Calculus is the branch of calculus that deals with integrals and their properties. Integrals are used to compute areas, volumes, central points, and many useful things. The two main types of integrals are definite and indefinite integrals. Key Concepts

1. Indefinite Integrals - Definition : The indefinite integral of a function f(x) is a function F(x) such that F′(x) = f(x). - Notation : ∫ f(x) dx = F(x) + C, where C is the constant of integration. 2. Definite Integrals - Definition : The definite integral of a function f(x) from a to b gives the area under the curve of f(x) from x = a to x = b. - Notation : (^) ∫ 𝑓(𝑥) 𝑏 𝑎 dx - Fundamental Theorem of Calculus : o Part 1: If F(x) is an antiderivative of f(x), then (^) ∫ 𝑓(𝑥) 𝑏 𝑎 dx = F(b) − F(a). o Part 2: 𝑑 𝑑𝑥

𝑥 𝑎 dt = f(x).

3. Basic Rules of Integration - Power Rule : ∫ xn^ dx = 𝑥𝑛+^1 𝑛+ 1 + C (for n ≠ - 1) - Constant Rule : ∫ c dx = cx + C - Sum/Difference Rule : ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫g(x) dx

3. Evaluate at the bounds x = 0 and x = 2 : o F(x) = 2x^2 − x o F(2) = 2(2)^2 − 2 = 8 − 2 = 6 o F(0)=2(0)^2 – 0 = 0
4. Compute F(2) − F(0): o 6 − 0 = 6
5. The value of the definite integral is 6. 3. Using Substitution Problem : Evaluate the integral ∫ (2x) (x^2 + 1)^3 dx. Solution :
6. Use substitution. Let u = x^2 + 1. Then du = 2x dx.
7. Rewrite the integral in terms of u: o ∫ (2x) (x2 + 1 )^3 dx = ∫ u^3 du
8. Integrate using the power rule: o ∫ u^3 du = 𝑢^3 +^1 3 + 1

+ C =

𝑢^4 4

+ C

4. Substitute uuu back in terms of x: o (𝑥^2 + 1 )^4 4

+ C

Conclusion

Integral calculus involves finding antiderivatives and evaluating the area under curves. By mastering the fundamental rules and techniques of integration, you can solve a wide range of problems. Practice with various types of integrals to solidify your understanding and improve your skills. Practice Problems

1. Find the indefinite integral ∫ (5x^4 − 2x^3 + x − 7) dx.
2. Evaluate the definite integral ∫ ( 3 x^2 − 4x + 2 ) 3 1 dx.
3. Use integration by parts to evaluate ∫ xex^ dx.
4. Use substitution to evaluate ∫ cos(3x) dx.