Antiderivatives and Indefinite Integrals — Basic Exercises

1Problem 1

Evaluate $$\int (3x^2-2)dx$$\int(3x^2-2)\,dx.

1.1Sample Solution

The result is $$\{x\mapsto x^3-2x+C\mid C\in \mathbb{R}\}$$\{x\mapsto x^3-2x+C\mid C\in\mathbb R\}. Differentiation verifies that $$(x^3-2x+C{)}^{\prime }=3x^2-2$$(x^3-2x+C)'=3x^2-2.

1.2Explanation

An indefinite integral is not a single function but the family of all antiderivatives.

2Problem 2

Let $F:\mathbb{R}\to \mathbb{R}$F:\mathbb R\to\mathbb R. Find $F$F satisfying $$F^{\prime }\left(x\right)=2x,F\left(1\right)=5.$$F'(x)=2x,\qquad F(1)=5.

2.1Sample Solution

Write $$F\left(x\right)=x^2+C$$F(x)=x^2+C. The condition gives $$F\left(1\right)=1+C=5$$F(1)=1+C=5, so $C=4$C=4 and therefore $$F\left(x\right)=x^2+4$$F(x)=x^2+4.

2.2Explanation

An additional condition determines the constant of integration.

3Problem 3

Explain the types of output produced by $${\int }_0^{1}2xdx$$\int_0^1 2x\,dx and $$\int 2xdx$$\int 2x\,dx.

3.1Sample Solution

The expression $${\int }_0^{1}2xdx$$\int_0^1 2x\,dx is a definite integral with specified endpoints, so its output is a single number. The expression $$\int 2xdx$$\int 2x\,dx is an indefinite integral, so its output is the family of functions $$\{x\mapsto x^2+C\mid C\in \mathbb{R}\}$$\{x\mapsto x^2+C\mid C\in\mathbb R\}.

3.2Explanation

Although the notation is related, definite and indefinite integrals have different definitions and different types of output.

4Related Material