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Chapter 4: Problem 43

Find each integral. $$ \int\left(\frac{3}{x}-5 e^{2 x}+\sqrt{x^{7}}\right) d x, x>0 $$

Short Answer

Expert verified
The integral is \(3\ln|x| - \frac{5}{2}e^{2x} + \frac{2}{9}x^{9/2} + C\).

Step by step solution

01

Separate the Integral

The given integral is a sum of three different functions integrated with respect to \(x\): \(\int \left( \frac{3}{x} - 5e^{2x} + \sqrt{x^7} \right) dx\). We can separate the integral into three distinct parts: \(\int \frac{3}{x} \, dx\), \(- \int 5e^{2x} \, dx\), and \(\int \sqrt{x^7} \, dx\).
02

Integrate \(\int \frac{3}{x} \ dx\)

Recognize that \(\int \frac{3}{x} \, dx\) is a simple logarithmic function. The integral of \(\frac{1}{x}\) is \(\ln|x|\), so \(\int \frac{3}{x} \, dx = 3\ln|x| + C_1\), where \(C_1\) is a constant of integration.
03

Integrate \(- \int 5e^{2x} \ dx\)

We need to integrate the exponential \(5e^{2x}\). Use the substitution \(u = 2x\), with \(du = 2 \, dx\) or \(dx = \frac{1}{2}du\). The integral becomes \(-5 \cdot \frac{1}{2} \int e^u du = -\frac{5}{2}e^{2x} + C_2\).
04

Integrate \(\int \sqrt{x^7} \ dx\)

Convert \(\sqrt{x^7}\) to \(x^{7/2}\). The integral \(\int x^{7/2} \, dx\) can be computed using the power rule: \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \(n = \frac{7}{2}\). This gives \(\frac{x^{9/2}}{9/2} = \frac{2}{9}x^{9/2} + C_3\).
05

Combine All Parts Together

Add the results from each integral to get the final answer: \(3\ln|x| - \frac{5}{2}e^{2x} + \frac{2}{9}x^{9/2} + C\), where \(C = C_1 + C_2 + C_3\) is the constant of integration.

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Key Concepts

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

Logarithmic Integration
Logarithmic integration is crucial when the integrand involves a term like \(\frac{1}{x}\). This type of function is linked to the natural logarithm function. The integral of \(\frac{1}{x}\) with respect to \(x\) results in \(\ln|x|\). Therefore, if your integrand includes constants such as \(\frac{3}{x}\), like in our exercise, simply multiply this constant with the result.Always remember:
  • The absolute value is essential because logarithms of non-positive numbers are undefined in the real number system.
  • Add a constant of integration to account for the family of antiderivatives, represented as \(C\).
Understanding this helps in recognizing and solving integrals of similar forms quickly.
Exponential Integration
Exponential integration often involves functions that include exponentials such as \(e^{kx}\). These are straightforward yet important for calculus problems. The standard secret weapon here is substitution. For example, consider integrals involving terms like \(5e^{2x}\). The process involves:
  • Substitute \(u = 2x\) and find \(du = 2 \, dx\), rearranging gives \(dx = \frac{1}{2} du\).
  • The integral then simplifies to \(\frac{1}{2} \int e^{u} du\) and evaluates to \(\frac{1}{2} e^u\).
  • Finally, substitute back to get \(\frac{1}{2} e^{2x}\).
This method provides a systematic approach to tackling exponentials by converting them into a more manageable form before integrating.
Power Rule Integration
The power rule is a fundamental concept in calculus, particularly for polynomial-like expressions. If your integrand is in the form of \(x^n\), where \(n\) is a constant, the power rule applies directly.For example, use the power rule of integration:
  • The integral of \(x^n\) with respect to \(x\) is \(\frac{x^{n+1}}{n+1} + C\).
  • Consider \(\sqrt{x^7}\) as \(x^{7/2}\). Using the power rule here involves noticing that the power increases by one. So, the integral becomes \(\frac{x^{9/2}}{9/2}\).
  • This simplifies quickly to \(\frac{2}{9}x^{9/2}\), remembering to incorporate the constant of integration \(C\).
Understanding the power rule is instrumental for efficiently managing polynomials and their variations like \(\sqrt{x}\).

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