Question

Find the indefinite integral. $$\int x^{3} d x$$

   Find the indefinite integral.
$$\int x^{3} d x$$

Applied Calculus for the MLSS A Brief Approach

Applied Calculus for the MLSS A Brief Approach

Soo T. Tan 10th Edition

Chapter 6, Problem 11

Instant Answer

Solved by Expert Nishant Tyagi

05/06/2021

Step 1

The power rule states that the integral of $x^n$ is $\frac{1}{n+1}x^{n+1}$, where $n$ is any real number except for -1. In this case, we have $n=3$, so the integral of $x^3$ is: $$\int x^3 dx = \frac{1}{3+1}x^{3+1} + C$$ Show more…

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Find the indefinite integral. $$\int x^{3} d x$$

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Key Concepts

Constant of Integration

The constant of integration is included in the result of an indefinite integral because antiderivatives are determined up to an arbitrary constant. When integrating, any function differing by a constant will still have the same derivative, which is why this constant, usually denoted as C, is added to the solution.

Power Rule for Integration

The power rule for integration is a technique used to find the antiderivative of power functions. It states that the antiderivative of x^n is (x^(n+1))/(n+1) provided that n is not equal to -1. This rule provides a straightforward method for integrating polynomial functions, making it a fundamental tool in calculus.

Indefinite Integral

An indefinite integral represents the collection of all antiderivatives of a function. It is expressed without specified limits of integration and includes an arbitrary constant, because the derivative of any constant is zero. This concept is central to reverse differentiation processes in calculus.

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