What is the integral of $$x^n$$ where @@n eq -1@@?

The integral of $$x^n$$ is given by $$\frac{x^{n+1}}{n+1} + C$$ , where $$C$$ is the constant of integration.

Integrate $$\frac{1}{x^2}$$ .

The integral of $$\frac{1}{x^2}dx$$ is $$-x^{-1} + C$$ .

Integrate $$\frac{6}{x^3}$$ .

The integral of $$\frac{6}{x^3}dx$$ is $$-3x^{-2} + C$$ .

Integrate $$8x^{-3}$$ .

The integral of $$8x^{-3}dx$$ is $$-4x^{-2} + C$$ .

Integrate $$5x^2 - 7x + 6$$ .

The integral of $$5x^2 - 7x + 6dx$$ is $$\frac{5}{3}x^3 - \frac{7}{2}x^2 + 6x + C$$ .

What is the power rule for integration?

The power rule states that for any real number @@n eq -1 $$, the integral of $$ x^n $$ is $$ rac{x^{n+1}}{n+1} + C@@.

Integrate $$\frac{1}{x^3} + \frac{1}{x^2}$$ .

The integral of $$\frac{1}{x^3} + \frac{1}{x^2}dx$$ is $$-\frac{1}{2}x^{-2} - x^{-1} + C$$ .

What is the integral of a constant $$k$$ ?

The integral of a constant $$k$$ is $$kx + C$$ .

Explain the difference between definite and indefinite integrals.

Indefinite integrals represent a family of functions and include a constant of integration $$C$$ , while definite integrals calculate the area under the curve between two specific limits and do not include $$C$$ .

Integrate $$x^{\frac{1}{3}}$$ .

The integral of $$x^{\frac{1}{3}}dx$$ is $$\frac{3}{4}x^{\frac{4}{3}} + C$$ .

What is the integral of $$x^{-1}$$ ?

The integral of $$x^{-1}dx$$ is $$\text{ln}|x| + C$$ .

Integrate $$\frac{1}{x^2} + \frac{6}{x^3}$$ .

The integral of $$\frac{1}{x^2} + \frac{6}{x^3}dx$$ is $$-x^{-1} - 3x^{-2} + C$$ .

What is the significance of the integral in real-world applications?

Integrals are used to calculate areas, volumes, and other quantities that accumulate over a continuous range, such as distance traveled over time or total profit over a period.

Basic Integration Flashcards

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1.

FLASHCARD QUESTION

Front

Define integration in calculus.

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Integration is the process of finding the integral of a function, which represents the area under the curve of that function on a graph.

2.

FLASHCARD QUESTION

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3.

FLASHCARD QUESTION

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What is the constant of integration?

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4.

FLASHCARD QUESTION

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5.

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6.

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