Calculus Home

Single variable calculus
1. Limits of sequences
2. Limits of functions. Continuity
3. Derivatives
4. Basic properties of derivatives
5. Chain rule
6. Maximum and minimum of a function
7. L'Hopital's Theorem
8. Antiderivatives and indefinite integrals
9. Definite integrals. Fundamental theorem of calculus
10. Substitution
11. Integration by parts
12. Integration of rational functions
13. Integration of trigonometric functions
14. Trigonometric substitutions
Multivariable calculus
15. Vectors. Equations of lines and planes
16. Functions of several variables
17. Vector fields. Parametric equations of curves and surfaces
18. Partial derivatives
19. Tangent planes
20. Gradient vector
21. Chain rule
22. Lagrange multipliers
23. Double integrals
24. Triple integrals
25. Change of variables in multiple integrals
26. Line integrals
27. Surface integrals
28. Curl and divergence
29. Green’s theorem, divergence theorem, and Stokes’ theorem
30. Formulas and summary

Antiderivatives and Indefinite Integrals

Definition

Definition (Antiderivative) A function \(F\) is an antiderivative of a function \(f\) on the interval \((a,b)\) if \(F^{\prime}(x)=f(x)\) for every \(x\in(a,b)\).

The function \(f(x)=3x^2\) has many anti-derivatives: \(F_1(x)=x^3\), \(F_2(x)=x^3-17\), \(F_3(x)=x^3+41\), etc. They all differ by a constant factor.

Definition (Indefinite integral) The set of all antiderivatives of a given function \(f\) is called the indefinite integral of \(f\), and it is denoted as \(\int f(x)\,dx\). More precisely \[\int f(x)\,dx=\left\{ F: F^{\prime}(x)=f(x)\right\}.\]

Consider the function \(f(x)=\cos x\). For each real number \(C\), the function \(F_C(x)=\sin x+C\) is an antiderivative of \(f\). We write \[\int \cos x\,dx=\sin x+C.\]

Main properties of indefinite integrals

Theorem If \(F_1\) and \(F_2\) are two antiderivatives of \(f\) then there exists a real number \(C\) such that \(F_1(x)-F_2(x)=C\) for all \(x\).

The following theorem is easy to prove using the main properties of derivatives.

Theorem

For any function \(f\) and any real number \(\alpha\) the following identity holds: \[\int \alpha \cdot f(x)\,dx=\alpha\int f(x)\,dx.\]

For any two functions \(f\) and \(g\) the following identity holds: \[\int (f+g)(x)\,dx=\int f(x)\,dx+\int g(x)\,dx.\]

We can use the previous theorem to find anti-derivatives of polynomials. For example, if \(P(x)=x^3-2x^2+11x+4\), then \[\int P(x)\,dx=\int x^3\,dx-2\int x^2\,dx+11\int x\,dx+4\int 1\,dx= \frac14x^3-2\frac{x^3}3+11\frac{x^2}2+4x+C,\] where \(C\) could be any real number.

Practice problems

Problem 1. Let \[F(x)=\cos^2 x-\sin^2 x,\] \[G(x)=2\cos^2x,\] \[H(x)=2\sin^2x,\] \[K(x)=\cos (2x).\] Which of the previously defined functions are anti-derivatives of the function \[\varphi(x)=-4\sin x\cdot\cos x.\]
  • (A) \(F\) only
  • (B) \(F\) and \(K\) only
  • (C) \(F\), \(G\), and \(K\) only
  • (D) \(H\) and \(K\) only
  • (E) \(H\) only

Problem 2. Determine \[\int \left(x^3-3\cos x\right)\,dx.\]

Problem 3. Determine \[\int\frac1{\cos^2x}\,dx.\]
  • (A) \(\ln(\cos x)\)
  • (B) \(\ln(\cos x)+C\)
  • (C) \(\ln(\cos x + C)\)
  • (D) \(\tan x\)
  • (E) \(\tan x+C\)

Problem 4. Assume that \(p\) is a real number different from \(-1\). Find \[\int x^p\,dx.\]
  • (A) \(\frac1{p+1}x^{p+1}\)
  • (B) \(px^{p-1}\)
  • (C) \(0\)
  • (D) \(\frac1{p+1}x^{p+1}+C\)
  • (E) \(px^{p-1}+C\)

Problem 5. Determine \[\int\frac1{x^2+1}\,dx.\]
  • (A) \(\arctan x+C\)
  • (B) \(\arctan x\)
  • (C) \(\ln(1+x^2)+C\)
  • (D) \(\ln(1+x^2)\)
  • (E) \(x\ln (1+x^2)\)