Definite integral

Function, Its Antiderivative, and Definite Integral

Submitted by michaela.bailova on Fri, 09/13/2024 - 16:53

2010013808

Level:

Evaluate the following definite integral.

\int_{- 1}^{0} \sqrt[4]{x^{4}} d x

\frac{1}{2}

0

- \frac{1}{2}

This integral cannot be evaluated.

2010013807

Level:

Evaluate the following definite integral.

\int_{- 2}^{- 1} \sqrt{x^{2}} d x

\frac{3}{2}

0

- \frac{3}{2}

This integral cannot be evaluated.

2010013806

Level:

Let

sgn (x) = {\begin{cases} 1, & x > 0 \\ 0, & x = 0. \\ - 1, & x < 0 \end{cases}

Evaluate the following definite integral.

\int_{- 3}^{- 2} (sgn (x + 1) - 1) d x

- 2

0

- 1

This integral cannot be evaluated.

2010013805

Level:

Let

sgn (x) = {\begin{cases} 1, & x > 0 \\ 0, & x = 0. \\ - 1, & x < 0 \end{cases}

Evaluate the following definite integral.

\int_{- 2}^{- 1} (sgn (x - 1) + 1) d x

0

- 1

2

This integral cannot be evaluated.

2010013804

Level:

Any positive real number

x

can be written as

x = c + d

, where

c

is an integer and

d \in [0, 1)

. Then

c

is called the integer part of

x

and is denoted by

[x]

. Evaluate the following definite integral.

\int_{3.1}^{\frac{7}{2}} [x] d x

1.2

1.6

3

This integral cannot be evaluated.

2010013803

Level:

Any positive real number

x

can be written as

x = c + d

, where

c

is an integer and

d \in [0, 1)

. Then

c

is called the integer part of

x

and is denoted by

[x]

. Evaluate the following definite integral.

\int_{\frac{5}{2}}^{2.8} [x] d x

0.6

0.9

2

This integral cannot be evaluated.

2010013802

Level:

Evaluate the following definite integral.

\int_{- 2}^{0} (1 - | x + 1 |) d x

1

2

- 2

This integral cannot be evaluated.

2010013801

Level:

Evaluate the following definite integral.

\int_{0}^{2} (| x - 1 | + 1) d x

3

2

- 2

This integral cannot be evaluated.

2010008006

Level:

Compare the two definite integrals

I_{1} = \int_{0}^{1} (x^{3} - x) d x

and

I_{2} = \int_{1}^{0} (x - x^{3}) d x

I_{1} = I_{2}

I_{1} > I_{2}

I_{1} < I_{2}

These integrals cannot be compared.

Function, Its Antiderivative, and Definite Integral

2010013808

2010013807

2010013806

2010013805

2010013804

2010013803

2010013802

2010013801

2010008006

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