# 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 function \(g(x)=F_1(x)-F_2(x)\) satisfies \(g^{\prime}(x)=0\) for all \(x\). Therefore \(g\) must be a constant function.

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

It is easy to verify that \(F^{\prime}(x)=G^{\prime}(x)=K^{\prime}(x)=-2\sin x\cdot \cos x=\varphi(x)\). The derivative of \(H\) is \(H'(x)=-4\sin x\cos x\). Hence, the answer is C.

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

We use the basic properties of the indefinite integral to obtain:
\[\int \left(x^3-3\cos x\right)\,dx=\int x^3\,dx-3\int \cos x\,dx=\frac{x^4}4-3\sin x+C.\]

**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)** \(\tan x+C\)

Direct verification shows that the answer is E.

**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)** \(px^{p-1}+C\)

Direct verification shows that the derivative of \(F(x)=\frac1{p+1}x^{p+1}+C\) is equal to \(x^p\). The correct answer is \(D\).

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

Direct verification shows that the answer is A.