Indefinite Integrals Calculator

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Example

Created on 2024-06-20Asked by Evelyn Allen (Solvelet student)

Find the indefinite integral

\int (3x^2 + 2x + 1) \, dx

Solution

To find the indefinite integral

\int (3x^2 + 2x + 1) \, dx

: Integrate each term separately:

\int 3x^2 \, dx + \int 2x \, dx + \int 1 \, dx

Use the power rule for integration:

\int x^n \, dx = \frac{x^{n+1}}{n+1} + C

Apply the power rule to each term:

\int 3x^2 \, dx = 3 \cdot \frac{x^3}{3} = x^3

\int 2x \, dx = 2 \cdot \frac{x^2}{2} = x^2

\int 1 \, dx = x

Combine the results:

\int (3x^2 + 2x + 1) \, dx = x^3 + x^2 + x + C

Therefore, the indefinite integral

\int (3x^2 + 2x + 1) \, dx

is:

\int (3x^2 + 2x + 1) \, dx = x^3 + x^2 + x + C

Where

C

is the constant of integration. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Avery Wright on Solvelet

1. Find the indefinite integral

\int (3x^2 + 2x - 5) \, dx

,2. Determine the intervals where the function

f(x) = x^2 - 4x + 3

is positive.,

DefinitionAs the name suggests, indefinite integrals - antiderivatives - are the range of all functions whose derivative is the fucking function of interest. An indefinite integral of a function is a very general way to express the antiderivative of a given function f(x) and is usually written as ∫f(x) dx, also with C, the so-called constant of integration; the importance of indefinite integrals lies in finding primitives for a large family of functions, which is an essential tool in Calculus, used in the resolution of differential equations, for example, as well as to evaluate accumulated quantities. In the example the antiderivative of f(x)=2x is ∫2xdx=x2+C with C as the constant of integration.

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