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Chapter 11: Problem 13

Evaluate the definite integral. $$ \int_{0}^{\pi / 2} \frac{d x}{5 \sin x+3} $$

Short Answer

Expert verified
\( \frac{1}{5} \ln \left( \frac{8}{3} \right) \)

Step by step solution

01

Express the Integrand

Write the definite integral as oindent \ oindent \[\int_{0}^{\pi / 2} \frac{d x}{5 \sin x+3}\]
02

Substitution

Use the substitution method to simplify the integral. Let \[ sin x = t \Rightarrow d(\sin x) = \cos x \, dx \Rightarrow dx = \frac{dt}{\cos x}\]Change the limits accordingly: when \[ x = 0, t = \sin(0) = 0\] and when \[x = \frac{\pi}{2}, t = \sin(\frac{\pi}{2})=1\]
03

Simplified Integral

Rewrite the integral with the new variable \[ \int_{0}^{1} \frac{dt}{5t+3}\]
04

Integration

Integrate the function using the result for the integral of the form \(\int \frac{1}{ax+b} \, dx \). We get: \[ \int \frac{1}{5t+3} \, dt = \frac{1}{5} \ln|5t+3| + C \] Evaluate within the new limits [0, 1].
05

Back Substitution and Evaluation

Evaluate the definite integral using the original limits:\[ \left. \frac{1}{5} \ln|5t+3| \right|_{0}^{1} = \left( \frac{1}{5} \ln|5 \cdot 1 + 3| \right) - \left( \frac{1}{5} \ln|5 \cdot 0 + 3| \right) = \frac{1}{5} \ln 8 - \frac{1}{5} \ln 3 \]Use logarithm properties to combine the subtraction into a fraction: \[ \frac{1}{5} \ln \left( \frac{8}{3} \right) \]

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Key Concepts

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

Substitution Method
When faced with complicated integrals, substitution is a powerful technique to simplify them. The key idea is to find a substitution that makes the integral easier to evaluate.
For our integral \(\frac{dx}{5 \sin x + 3}\), we used \(t = \sin x\).
This converts our integral into a function of \(t\), which is easier to integrate.
By changing variables, we can handle the integral in a more straightforward manner.
Integration Techniques
Integration is a fundamental concept in calculus, often used to find areas under curves.
Different techniques are required for different types of functions. Here, we used substitution, but other methods include:
  • Partial Fractions
  • By Parts
  • Trigonometric Integrals
Learning a variety of techniques equips you to tackle a plethora of integrative challenges. Mastery comes from practice and recognizing patterns in different functions.
Logarithmic Integration
Logarithmic integration helps solve integrals of the form \(\frac{1}{ax+b}\).
In our problem, we arrived at \(\int \frac{1}{5t+3} \, dt\) and used its direct integral:
\(\frac{1}{a} \ln |ax + b| + C\).
This property makes logarithmic integration straightforward in cases involving linear denominators.
Definite Integrals
Definite integrals provide the net area under a curve between two points. Unlike indefinite integrals, they result in a numerical value.
After performing the integration, replace the variable back and evaluate from the lower limit to the upper limit:
\(\frac{1}{5} \ln \left| 8 \right/ 3 \right|\), simplified using properties of logarithms.
The process involves:
  • Changing variable limits
  • Integrating
  • Evaluating within those limits
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Most popular questions from this chapter

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