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