Problem 57
Question
In Exercises \(53-66,\) make a \(u\) -substitution and integrate from \(u(a)\) to \(u(b) .\) $$\int_{0}^{1} \frac{10 \sqrt{\theta}}{\left(1+\theta^{3 / 2}\right)^{2}} d \theta$$
Step-by-Step Solution
Verified Answer
The result of the integral, using u-substitution method, is \(\frac{10}{3}\).
1Step 1: Choose the Appropriate Substitution
Choose \(u\) as \[u = 1+ \theta^{3/2}\].
2Step 2: Find the Derivative of \(u\)
Differentiate \(u\) with respect to \( \theta \): \[du = \frac{3}{2} \theta^{1/2} d \theta\]. Multiply both sides by \(\frac{2}{3}\) to get \[\frac{2}{3} du = \theta^{1/2} d \theta\].
3Step 3: Substitute \(u\) and \(du\) into the Integral
Substitute \(u\) and \(du\) into the integral to simplify it: \[\int \frac{10 \sqrt{\theta}}{(u)^{2}} \cdot \frac{2}{3} du\]. After simplifying, we get: \[\frac{20}{3} \int \frac{1}{u^2} du \].
4Step 4: Change the Limits of Integration
Change the limits of integration accordingly. For \(\theta = 0\), calculate \(u(0)\), which gives \(u = 1+0^{3/2} = 1 \). Similarly, for \(\theta = 1\), compute \(u(1)\), which results in \(u = 1+1^{3/2} = 2\). Therefore, the new limits are from 1 to 2.
5Step 5: Apply the Power Rule of Integration
Determine the antiderivative by applying the power rule of integration: \[\frac{20}{3} \int_{1}^{2} u^{-2} du = \frac{20}{3} [- u^{-1} ]_{1}^{2}\]. After evaluating from 1 to 2, we get \[= \frac{20}{3} [-1/2 - (-1)] = \frac{20}{3} * (1/2) = \frac{20}{6} =\frac{10}{3} .\]
6Step 6: State the Final Answer
After evaluating the integral using the \(u\)-substitution method and simplifying the expression, the final answer is \(\frac{10}{3}\).
Key Concepts
Definite IntegralPower Rule of IntegrationIntegration by SubstitutionChanging Limits of Integration
Definite Integral
The definite integral is a fundamental concept in calculus, representing the signed area under a curve bounded by specific limits, usually interpreted as the accumulated total value of a rate function. When performing a definite integral, we calculate the integral between two points: the lower limit 'a' and the upper limit 'b'. Mathematically, it's expressed as \(\int_{a}^{b} f(x) dx\).
In the context of the problem, \(\int_{0}^{1} \frac{10 \sqrt{\theta}}{\left(1+\theta^{3 / 2}\right)^{2}} d\theta\) asked us to find the area under the curve of the given function between \(\theta = 0\) and \(\theta = 1\). This process can often be simplified through the use of various integration techniques, one of which is u-substitution.
In the context of the problem, \(\int_{0}^{1} \frac{10 \sqrt{\theta}}{\left(1+\theta^{3 / 2}\right)^{2}} d\theta\) asked us to find the area under the curve of the given function between \(\theta = 0\) and \(\theta = 1\). This process can often be simplified through the use of various integration techniques, one of which is u-substitution.
Power Rule of Integration
The power rule of integration is one of the most essential rules in calculus, enabling us to integrate functions of the form \(x^n\). The rule states that, if \(n eq -1\), the integral of \(x^n\) with respect to x is \(\frac{1}{n+1}x^{n+1}\) plus the constant of integration 'C' in the indefinite case. This rule greatly simplifies the integration of polynomial functions.
During the process of solving the integral in our exercise, we arrive at a point where we can apply the power rule after performing a u-substitution. The power rule effectively provides us with the antiderivative of \(u^{-2}\), which is crucial to finding the definite integral's value.
During the process of solving the integral in our exercise, we arrive at a point where we can apply the power rule after performing a u-substitution. The power rule effectively provides us with the antiderivative of \(u^{-2}\), which is crucial to finding the definite integral's value.
Integration by Substitution
Integration by substitution, also known as u-substitution, is a technique used to simplify integrals. The goal is to transform the integral into a basic form, where we can easily apply the power rule or other easy integration techniques. We achieve this by substituting a part of the integral with a new variable 'u' and its derivative 'du'. The choice of 'u' is vital and usually involves recognising a function whose derivative also appears in the integral.
In our example, we picked \(u = 1+ \theta^{3/2}\) due to the appearance of its derivative \(\frac{3}{2} \theta^{1/2} d \theta\) in the integral. This strategic choice transformed the complicated original integral into a form where we could readily apply the integration power rule.
In our example, we picked \(u = 1+ \theta^{3/2}\) due to the appearance of its derivative \(\frac{3}{2} \theta^{1/2} d \theta\) in the integral. This strategic choice transformed the complicated original integral into a form where we could readily apply the integration power rule.
Changing Limits of Integration
When we perform a u-substitution in a definite integral, the limits of integration must also change accordingly. Originally, these limits were defined with respect to the variable 'x' (or in the case of our problem, \(\theta\)). However, once we transition into the u-variable space, we need to recalculate these limits so that they correspond to the values of 'u' at the endpoints.
For instance, after substituting in our problem, the original limits \(0\) to \(1\) denoting the \(\theta\)-range were replaced by \(1\) to \(2\), which correspond to the range of 'u'. This is done to accurately capture the interval over which we're integrating in the u-space. Recalculating the limits is critical because using the original \(\theta\)-limits after substituting would render the integral incorrect or meaningless.
For instance, after substituting in our problem, the original limits \(0\) to \(1\) denoting the \(\theta\)-range were replaced by \(1\) to \(2\), which correspond to the range of 'u'. This is done to accurately capture the interval over which we're integrating in the u-space. Recalculating the limits is critical because using the original \(\theta\)-limits after substituting would render the integral incorrect or meaningless.
Other exercises in this chapter
Problem 56
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In Exercises \(53-66,\) make a \(u\) -substitution and integrate from \(u(a)\) to \(u(b) .\) $$\int_{-\pi}^{\pi} \frac{\cos x}{\sqrt{4+3 \sin x}} d x$$
View solution