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Chapter 8: Problem 33

Find the indefinite integral. $$ \int e^{5 x} d x $$

Short Answer

Expert verified
The indefinite integral of \(e^{5x}\) is \(\frac{1}{5} e^{5x} + C\).

Step by step solution

01

Identify the Integral to Solve

Firstly, the given function to be integrated can be expressed as \(e^{5x}\). Here, the goal is to find its indefinite integral.
02

Apply the Integration Rule for Exponential Functions

The integral of \(e^{nx}\) is \(1/n e^{nx}\). Hence, replacing n with 5 (the coefficient of x in the exponent), the indefinite integral becomes \(\frac{1}{5} e^{5x} + C\), where C refers to the constant of integration which arises from the fundamental theorem of calculus.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Functions
Exponential functions are mathematical representations where the variable appears as an exponent. A classic example is the function \(e^{x}\), where \(e\) is the base of the natural logarithm, approximately equal to 2.718. Exponential functions exhibit continuous and rapid growth or decay, making them crucial in various mathematical and real-world applications.
In the function \(e^{5x}\), the base \(e\) is raised to the power of \(5x\). The coefficient 5, multiplying the variable \(x\), influences the rate of increase or decrease of the function. When integrating exponential functions, recognizing the specific form \(e^{nx}\) helps in applying the correct integration methods.
Integration Rule
The integration rule for exponential functions is a pivotal tool in calculus. Specifically, when finding the indefinite integral of functions like \(e^{nx}\), the rule states that the integral is \(\frac{1}{n} e^{nx}\). This involves dividing the exponential function by the coefficient \(n\) of the exponent.
  • First, identify the coefficient \(n\) of the variable \(x\) in the exponent. For \(e^{5x}\), \(n = 5\).
  • Apply the rule: the integral of \(e^{5x}\) becomes \(\frac{1}{5} e^{5x}\).
This formula is derived from the inverse of the differentiation rule of exponential functions, making it efficient to compute integrals of similar types effortlessly.
Constant of Integration
Whenever you calculate an indefinite integral, you must also include a constant of integration, commonly denoted as \(C\). This constant represents the family of all possible antiderivatives.
Because the process of integration is essentially the reverse of differentiation, adding \(C\) accounts for the fact that differentiating a constant results in zero. The constant of integration signifies that various functions, differing only by a constant, can share the same derivative.
  • When solving \(\int e^{5x} dx\), it results in \(\frac{1}{5} e^{5x} + C\).
  • The \(+ C\) reminds us that without initial conditions or further information, there are countless functions which differ merely by some constant.
This ensures the solutions of integrals are as broad and inclusive as necessary within mathematical contexts.

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Most popular questions from this chapter

Show that \(\lim _{x \rightarrow \infty} \frac{x^{n}}{e^{x}}=0\) for any integer \(n>0\).

True or False? , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If the graph of \(f\) is symmetric with respect to the origin or the \(y\) -axis, then \(\int_{0}^{\infty} f(x) d x\) converges if and only if \(\int_{-\infty}^{\infty} f(x) d x\) converges.

True or False? , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f\) is continuous on \([0, \infty)\) and \(\lim _{x \rightarrow \infty} f(x)=0\), then \(\int_{0}^{\infty} f(x) d x\) converges.

The magnetic potential \(P\) at a point on the axis of a circular coil is given by $$ P=\frac{2 \pi N I r}{k} \int_{c}^{\infty} \frac{1}{\left(r^{2}+x^{2}\right)^{3 / 2}} d x $$ where \(N, I, r, k\), and \(c\) are constants. Find \(P\).

Find the value of \(c\) that makes the function continuous at \(x=0\). $$ f(x)=\left\\{\begin{array}{ll} \left(e^{x}+x\right)^{1 / x}, & x \neq 0 \\ c, & x=0 \end{array}\right. $$
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