Problem 294
Question
In the following exercises, use a change of variables to evaluate the definite integral. $$ \int_{0}^{2} \frac{t}{\sqrt{5+t^{2}}} d t $$
Step-by-Step Solution
Verified Answer
The integral evaluates to \(3 - \sqrt{5}\).
1Step 1: Identify the Substitution
Let's choose a substitution that will simplify the integrand. Notice that the term \(\sqrt{5+t^2}\) suggests a trigonometric substitution or direct algebraic substitution. We choose \( u = 5 + t^2 \) because differentiating gives us the term \(t dt\) in the integral.Then, differentiating both sides gives \( du = 2t \, dt \), which suggests that \( t dt = \frac{1}{2} du \).
2Step 2: Rewrite the Integral
Substitute \( t^2 = u - 5 \) and \( t dt = \frac{1}{2} du \) into the integral. Also, change the limits of integration by substituting the values of \( t \):- When \( t = 0 \), \( u = 5 + 0^2 = 5 \).- When \( t = 2 \), \( u = 5 + 2^2 = 5 + 4 = 9 \).Thus, the integral becomes \[ \int_{5}^{9} \frac{1}{\sqrt{u}} \cdot \frac{1}{2} \, du = \frac{1}{2} \int_{5}^{9} u^{-1/2} \, du. \]
3Step 3: Evaluate the Integral
Integrate \( \frac{1}{2} \int u^{-1/2} du \) using the power rule for integration. The antiderivative of \( u^{-1/2} \) is \( 2u^{1/2} \), so evaluate:\[ \frac{1}{2} \cdot \bigg[ 2u^{1/2} \bigg]_{5}^{9} = \bigg[ u^{1/2} \bigg]_{5}^{9}. \]
4Step 4: Evaluate the Definite Integral
Now substitute the upper and lower limits into the antiderivative:\[ [ u^{1/2}]_{5}^{9} = (9)^{1/2} - (5)^{1/2} = 3 - \sqrt{5}. \]
5Step 5: Write the Final Answer
Combine all parts to conclude that the original definite integral evaluates to:\[ \int_{0}^{2} \frac{t}{\sqrt{5+t^{2}}} \, dt = 3 - \sqrt{5}. \]
Key Concepts
Change of VariablesIntegration by SubstitutionLimits of IntegrationPower Rule for Integration
Change of Variables
When dealing with definite integrals, finding a simpler expression for an integrand can significantly ease the computation process. This is where the technique of change of variables comes into play. By replacing an element of the integral with a different variable, we simplify the integral into a more manageable form.
In the given example of \[ \int_{0}^{2} \frac{t}{\sqrt{5+t^{2}}} dt \], the original variable is \( t \). We introduce a new variable \( u \) such that \( u = 5 + t^2 \). This choice is strategic as it targets the complex square root term.
By making this substitution, we can transform complex parts of the integral into simpler algebraic forms. This makes integration easier and reduces the risk of errors, especially with complex expressions.
In the given example of \[ \int_{0}^{2} \frac{t}{\sqrt{5+t^{2}}} dt \], the original variable is \( t \). We introduce a new variable \( u \) such that \( u = 5 + t^2 \). This choice is strategic as it targets the complex square root term.
By making this substitution, we can transform complex parts of the integral into simpler algebraic forms. This makes integration easier and reduces the risk of errors, especially with complex expressions.
Integration by Substitution
Integration by substitution is a technique similar to the chain rule in differentiation. It helps us to integrate more complex functions by substituting part of the function with a single, more manageable variable.
In the case of our integral, \( \int_{0}^{2} \frac{t}{\sqrt{5+t^{2}}} dt \), after deciding \( u = 5 + t^2 \), we also derive \( du \) in terms of \( dt \): \( du = 2t \, dt \). Thus, \( t dt = \frac{1}{2} du \). By rewriting every part of the integral in terms of \( u \) and not \( t \), we effectively turn the problem into integrating \( \frac{1}{2} \int_{5}^{9} u^{-1/2} \, du \).
In the case of our integral, \( \int_{0}^{2} \frac{t}{\sqrt{5+t^{2}}} dt \), after deciding \( u = 5 + t^2 \), we also derive \( du \) in terms of \( dt \): \( du = 2t \, dt \). Thus, \( t dt = \frac{1}{2} du \). By rewriting every part of the integral in terms of \( u \) and not \( t \), we effectively turn the problem into integrating \( \frac{1}{2} \int_{5}^{9} u^{-1/2} \, du \).
- Identify part of the integrand that can be substituted to simplify the problem.
- Compute the derivative of the substitute function to transform \( dt \) or part of the integrand into terms of the new variable.
Limits of Integration
When we perform integration by substitution, especially in definite integrals, adjusting the limits of integration is crucial. The original integral had limits from 0 to 2.
However, since we've changed the variables from \( t \) to \( u \), the limits must also be recalculated in terms of \( u \).
However, since we've changed the variables from \( t \) to \( u \), the limits must also be recalculated in terms of \( u \).
- For \( t = 0 \), \( u = 5 + 0^2 = 5 \).
- For \( t = 2 \), \( u = 5 + 2^2 = 9 \).
Power Rule for Integration
The power rule is a fundamental integration rule that is quite handy for integrals of forms like \( u^n \). It states that \[ \int u^n \, du = \frac{u^{n+1}}{n+1} + C \] for \( n eq -1 \).
Applying it to our problem, once we've rewritten the integral as \( \frac{1}{2} \int_{5}^{9} u^{-1/2} \, du \), we can identify \( n = -1/2 \). Therefore, the antiderivative is \( 2u^{1/2} \) as integrating adds 1 to the exponent and divides by the new power, giving us \( \int u^{-1/2} \, du = 2u^{1/2} \).
This converts the integral to \[ \frac{1}{2} \cdot [2u^{1/2}]_{5}^{9} = [u^{1/2}]_{5}^{9}\]. The power rule simplifies calculating the integral evaluation step by offering a straightforward technique for addressing polynomial-like functions.
Applying it to our problem, once we've rewritten the integral as \( \frac{1}{2} \int_{5}^{9} u^{-1/2} \, du \), we can identify \( n = -1/2 \). Therefore, the antiderivative is \( 2u^{1/2} \) as integrating adds 1 to the exponent and divides by the new power, giving us \( \int u^{-1/2} \, du = 2u^{1/2} \).
This converts the integral to \[ \frac{1}{2} \cdot [2u^{1/2}]_{5}^{9} = [u^{1/2}]_{5}^{9}\]. The power rule simplifies calculating the integral evaluation step by offering a straightforward technique for addressing polynomial-like functions.
Other exercises in this chapter
Problem 292
In the following exercises, use a change of variables to evaluate the definite integral. $$ \int_{0}^{1} x \sqrt{1-x^{2}} d x $$
View solution Problem 293
In the following exercises, use a change of variables to evaluate the definite integral. $$ \int_{0}^{1} \frac{x}{\sqrt{1+x^{2}}} d x $$
View solution Problem 295
In the following exercises, use a change of variables to evaluate the definite integral. $$ \int_{0}^{1} \frac{t}{\sqrt{1+t^{3}}} d t $$
View solution Problem 296
In the following exercises, use a change of variables to evaluate the definite integral. $$ \int_{0}^{\pi / 4} \sec ^{2} \theta \tan \theta d \theta $$
View solution