Question

Find the following. $$\int\left(\frac{e}{x^{5}}+e x^{1 / 2}\right) d x$$

   Find the following.
$$\int\left(\frac{e}{x^{5}}+e x^{1 / 2}\right) d x$$

Calculus with Applications

Calculus with Applications

Margaret L. Lial •… 11th Edition

Chapter 7, Problem 23

Instant Answer

Solved by Expert Hayden Woerner

02/13/2020

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This gives us: $$\int\left(e x^{-5}+e x^{1 / 2}\right) d x$$ Show more…

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Find the following. $$\int\left(\frac{e}{x^{5}}+e x^{1 / 2}\right) d x$$

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Key Concepts

Constant of Integration

When performing indefinite integration, the result is not unique because any constant added to an antiderivative also satisfies the derivative. Therefore, an arbitrary constant, commonly denoted as C, is added to the final result. This constant represents the fact that there is a family of functions that differ only by a constant that all serve as valid antiderivatives.

Sum Rule in Integration

The sum rule for integration states that the integral of a sum of functions is equal to the sum of the integrals of the individual functions. This property is extremely useful because it allows you to separate complex integrals into simpler, more manageable parts, which can then be integrated independently.

Constant Multiple Rule

The constant multiple rule in integration states that a constant factor can be pulled out of the integral. This means if you have a constant multiplying a function, you can multiply the integral of that function by the constant. This rule is essential because it simplifies integration by allowing you to integrate the variable part separately from the constant multiplier.

Power Rule for Integration

The power rule for integration is a fundamental technique used to integrate functions of the form x^n, where n is any real number except -1. It states that the integral of x^n with respect to x is x^(n+1)/(n+1) plus an arbitrary constant. This rule simplifies the integration process for polynomial and power functions and is widely applicable in calculus.

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