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Magazine Chapter 5: Problem 12
Exer. 9-48: Evaluate the integral. $$ \int \frac{1}{\sqrt{4-5 t}} d t $$
Short Answer
Expert verified
\(-\frac{2}{5} \sqrt{4 - 5t} + C\)
Step by step solution
01
Identify the Integral
The integral given is \( \int \frac{1}{\sqrt{4-5t}} \, dt \). This is an indefinite integral that involves a square root in the denominator, suggesting the use of u-substitution for simplification.
02
Choose a Substitution
To simplify the integral, let \( u = 4 - 5t \). This implies that \( du = -5 \, dt \). Rearrange this to express \( dt \) in terms of \( du \): \( dt = \frac{du}{-5} \).
03
Substitute and Simplify
Substitute \( u = 4 - 5t \) and \( dt = \frac{du}{-5} \) into the integral:\[ \int \frac{1}{\sqrt{u}} \cdot \frac{du}{-5} = -\frac{1}{5} \int u^{-1/2} \, du \]
04
Integrate with Respect to \( u \)
The integral \( \int u^{-1/2} \, du \) can be found using the power rule for integration, \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \). Here, \( n = -1/2 \):\[ -\frac{1}{5} \cdot \frac{u^{1/2}}{1/2} + C = -\frac{1}{5} \cdot 2u^{1/2} + C = -\frac{2}{5} \sqrt{u} + C \]
05
Substitute Back to Terms of \( t \)
Replace \( u \) with \( 4 - 5t \) to get the solution in terms of \( t \):\[ -\frac{2}{5} \sqrt{4 - 5t} + C \]
06
Simplify the Expression
The final answer in terms of \( t \) is \[ -\frac{2}{5} \sqrt{4 - 5t} + C \] where \( C \) 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.
Indefinite Integral
An indefinite integral is often thought of as the opposite of taking a derivative. In more formal terms, it is the antiderivative of a function. When you calculate an indefinite integral, you're essentially finding a function whose derivative gives you back your original function.
- Unlike definite integrals, indefinite integrals do not have upper and lower bounds. They represent a general form of antiderivative.
- The result includes a constant of integration, denoted as \( C \), because when you take a derivative of a constant, it disappears, making the original constant unknown when working backwards.
U-Substitution
U-substitution is a powerful technique in integration that simplifies complex integrals by transforming them into basic forms. It is similar to reversing the chain rule used in differentiation.To perform u-substitution:
- First, identify a part of the integral that can be substituted with a simpler variable (let's call it \( u \)). This usually involves algebraic expressions inside functions or expressions that can be grouped together.
- After choosing \( u \), calculate \( du \), the differential of \( u \). This step helps rewrite the whole integral in terms of \( u \) and \( du \).
- Substitute back into the integral, replacing all original variables with \( u \) and \( du \). The integral should now have a simpler form that is easier to integrate directly.
Power Rule for Integration
The power rule for integration is one of the fundamental tools for finding antiderivatives. It states:\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]This rule applies when \( n eq -1 \).Here is how it works:
- Identify the exponent of the variable in the expression to be integrated.
- Increase the exponent by one and divide by the new exponent.
