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Chapter 11: Problem 12

Evaluate the definite integral. $$\int_{1}^{2}\left(x^{3}+\frac{3}{4}\right)\left(x^{4}+3 x\right)^{-2} d x$$

Short Answer

Expert verified
After using the substitution method with \(u = x^4 + 3x\), simplifying, and evaluating the integral, the value of the definite integral is \(\boxed{\frac{9}{80}}\).

Step by step solution

01

Identify the substitution

Let's choose the substitution \(u = x^4 + 3x\). This choice is motivated by the presence of the term \((x^4 + 3x)^{-2}\) in the integrand, which suggests that the substitution might simplify it.
02

Find du and the new limits

Now differentiate u with respect to x: \[du = \frac{d(u)}{dx} dx = (4x^3 + 3) dx\] We can rewrite this expression as: \[dx = \frac{du}{4x^3 + 3}\] Using our substitution, we also need to find the new limits of integration. When \(x = 1\): \[u = 1^4 + 3(1) = 4\] And when \(x = 2\): \[u = 2^4 + 3(2) = 20\] Now our new limits are \(u = 4\) up to \(u = 20\).
03

Substitute and simplify

Now substitute the expressions for u and dx to the integral: \[\int_{1}^{2}\left(x^{3}+\frac{3}{4}\right)\left(x^{4}+3 x\right)^{-2} dx = \int_4^{20} \left(x^3+ \frac{3}{4} \right)u^{-2} \cdot \frac{du}{4x^3 + 3}\] Note that \(4x^3\) is present in both the numerator and the denominator. So we can simplify the integral using our substitution: \[\int_4^{20} \left(\frac{u-3x}{4}+ \frac{3}{4} \right)u^{-2} \cdot \frac{du}{u-3x}\] The terms \(\frac{u-3x}{4}\) in the numerator and \(u-3x\) in the denominator cancel out, leaving us with: \[\int_4^{20} \frac{3}{4} u^{-2} du\]
04

Integrate and evaluate

Now we can integrate this simplified expression: \[\int_4^{20} \frac{3}{4} u^{-2} du = \frac{3}{4}\int_4^{20} u^{-2} du\] Using the power rule for integration, we obtain: \[\frac{3}{4}\left[-u^{-1}\right]_4^{20}\] Now we can evaluate the integral at the new limits of integration: \[\frac{3}{4}\left(-\frac{1}{20} + \frac{1}{4}\right) = \frac{3}{4}\left(\frac{3}{20}\right)\]
05

Simplify the result

Now simplify the result: \[\frac{9}{80}\] So the value of the definite integral is: \[\int_{1}^{2}\left(x^{3}+\frac{3}{4}\right)\left(x^{4}+3 x\right)^{-2} d x = \boxed{\frac{9}{80}}\]

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Key Concepts

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

Substitution Method
The substitution method is a powerful tool in calculus, especially useful for integrals. When evaluating a definite integral, the substitution method involves choosing a new variable to simplify the integrand, then rewriting the differential and limits of integration in terms of this new variable.

In this exercise, the substitution was selected as \( u = x^4 + 3x \). This choice is effective because it helps to transform the integral into a simpler form. By finding the derivative \( du = (4x^3 + 3) dx \), we express \( dx \) in terms of \( du \): \( dx = \frac{du}{4x^3 + 3} \).

Remember to also change the limits of integration according to the substitution, which leads to easier evaluation. This process is crucial because it transforms a complex integral into a more manageable one, as seen when the terms \( (x^4 + 3x)^{-2} \) are simplified into \( u^{-2} \).

Thus, the substitution method is a way of "rewording" the problem into something more solvable.
Limits of Integration
In definite integrals, the limits of integration describe the range over which the function is being integrated. When using the substitution method, it's essential to adjust these limits according to the new variable.

Originally, the problem had integration limits from \( x = 1 \) to \( x = 2 \). When substituting \( u = x^4 + 3x \), you must find the values of \( u \) at these points.
  • At \( x = 1 \), \( u = 1^4 + 3(1) = 4 \).
  • At \( x = 2 \), \( u = 2^4 + 3(2) = 20 \).

Thus, the new limits of integration become \( u = 4 \) to \( u = 20 \).

Altering limits in this way ensures that the new integral represents the same area under the curve as the original. It is an essential step to correctly evaluate the definite integral using substitution, as it maintains the boundary constraints of the original problem.
Power Rule for Integration
The power rule for integration is one of the fundamental rules applied when integrating polynomial functions. It is expressed as \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n \) is not equal to \(-1\). This rule aids in finding antiderivatives, transforming them into more familiar power expressions.

In this exercise, once the integral has been simplified to \( \int_4^{20} \frac{3}{4} u^{-2} \, du \), the power rule can be applied. The antiderivative of \( u^{-2} \) is found by increasing the power by one and dividing by the new power. Here:
  • The antiderivative of \( u^{-2} \) is \( -u^{-1} \).
  • Thus, \( \frac{3}{4} \int u^{-2} \, du = \frac{3}{4} \left[ -u^{-1} \right] \).

After integrating, the limits are evaluated to provide the definite integral's value.

This rule simplifies calculations and allows us to tackle more complex polynomial integrands with confidence.

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Most popular questions from this chapter

It is known that the quantity demanded of a certain make of portable hair dryer is \(x\) hundred units/week and the corresponding wholesale unit price is $$ p=\sqrt{225-5 x} $$ dollars. Determine the consumers' surplus if the wholesale market price is set at \(\$ 10 /\) unit.

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