Caluclus: Table of contents

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) \(x\ln (1+x^2)\)