Problem 7
Question
Evaluate the definite integral. $$\int_{0}^{1} \frac{1}{\sqrt{2 x+1}} d x$$
Step-by-Step Solution
Verified Answer
The short answer for evaluating the definite integral \(\int_{0}^{1} \frac{1}{\sqrt{2 x+1}} d x\) is \(\sqrt{3} - 1\).
1Step 1: Identify the substitution
Before integrating, let's simplify the integrand using a suitable substitution. Let:
\[u = 2x + 1\]
Then,
\[du = 2 dx\]
2Step 2: Rewrite integral in terms of u
Using the substitution, we rewrite the given integral as follows:
\[\frac{1}{2} \int_{0}^{1} \frac{1}{\sqrt{u}} du\]
3Step 3: Update the limits
Update the limits of integration according to the substitution:
\(u = 2(0) + 1 = 1\)
\(u = 2(1) + 1 = 3\)
So, the new integral becomes:
\[\frac{1}{2} \int_{1}^{3} \frac{1}{\sqrt{u}} du\]
4Step 4: Integrate
Now, integrate the new function with respect to u:
\[\frac{1}{2}\int_{1}^{3} u^{-1/2} du\]
Notice that this is in the form of a power rule for an integral.
To integrate this function, we will use the power rule for integration:
\[ \int x^n dx = \frac{x^{n+1}}{n+1} + C\]
So, in our case, n = -1/2:
\[\frac{1}{2} [\frac{u^{(-1/2)+1}}{(-1/2)+1}] |_1^3 = \frac{1}{2} [2u^{1/2}] |_1^3\]
5Step 5: Evaluate the integral within limits
Now, evaluate the expression within the given limits:
\[\frac{1}{2} \left[ 2(\sqrt{3} - \sqrt{1}) \right] = \sqrt{3} - 1\]
So, the definite integral evaluates to:
\[\int_{0}^{1} \frac{1}{\sqrt{2 x+1}} d x = \sqrt{3} - 1\]
Key Concepts
Substitution MethodPower Rule for IntegrationLimits of IntegrationIntegrand Simplification
Substitution Method
The substitution method is a technique used in integration to transform a difficult integral into a simpler one. This method involves substituting a part of the integrand (the function being integrated) with a new variable, usually denoted as \( u \). This simplifies the integral into a new form that is easier to evaluate.
For instance, in the original integral \( \int_{0}^{1} \frac{1}{\sqrt{2x+1}} \, dx \), the substitution \( u = 2x + 1 \) is chosen because the expression \( 2x + 1 \) is inside the square root. By setting \( u = 2x + 1 \), we tidy up the integrand's appearance and make the resulting integral more straightforward.
After declaring \( u = 2x + 1 \), we also need to express \( dx \) in terms of \( du \). Differentiating \( u \) with respect to \( x \) gives \( du = 2 \, dx \). Solving this for \( dx \), we get \( dx = \frac{1}{2} \, du \). This replacement completes the transformation, turning the integral into something easier to manage.
For instance, in the original integral \( \int_{0}^{1} \frac{1}{\sqrt{2x+1}} \, dx \), the substitution \( u = 2x + 1 \) is chosen because the expression \( 2x + 1 \) is inside the square root. By setting \( u = 2x + 1 \), we tidy up the integrand's appearance and make the resulting integral more straightforward.
After declaring \( u = 2x + 1 \), we also need to express \( dx \) in terms of \( du \). Differentiating \( u \) with respect to \( x \) gives \( du = 2 \, dx \). Solving this for \( dx \), we get \( dx = \frac{1}{2} \, du \). This replacement completes the transformation, turning the integral into something easier to manage.
Power Rule for Integration
The power rule for integration is a fundamental principle that tells us how to integrate expressions of the form \( x^n \). If \( n eq -1 \), the rule is:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
where \( C \) is the constant of integration.
In our exercise, after substitution, the integral becomes \( \int u^{-1/2} \, du \). Here, \( n = -1/2 \). Applying the power rule, we increase the exponent by 1, resulting in \( u^{1/2} \), and divide by the new exponent, giving us \( \frac{u^{1/2}}{1/2} \). When simplified, this becomes \( 2u^{1/2} \).
This systematic approach simplifies many integration problems, allowing us to find antiderivatives efficiently and with confidence.
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
where \( C \) is the constant of integration.
In our exercise, after substitution, the integral becomes \( \int u^{-1/2} \, du \). Here, \( n = -1/2 \). Applying the power rule, we increase the exponent by 1, resulting in \( u^{1/2} \), and divide by the new exponent, giving us \( \frac{u^{1/2}}{1/2} \). When simplified, this becomes \( 2u^{1/2} \).
This systematic approach simplifies many integration problems, allowing us to find antiderivatives efficiently and with confidence.
Limits of Integration
When performing a definite integral, it's crucial to adjust the limits of integration after substituting variables. This step ensures that the new variable reflects the original boundaries.
Initially, we are integrating from 0 to 1 in terms of \( x \). After substituting \( u = 2x + 1 \), we need to recalculate these limits for \( u \). Using the substitution variable, when \( x = 0 \), \( u = 2(0) + 1 = 1 \), and when \( x = 1 \), \( u = 2(1) + 1 = 3 \).
This change means that our rewritten integral now spans from \( u = 1 \) to \( u = 3 \), accurately reflecting the original integral's scope but in terms of \( u \) instead of \( x \). Ensuring correct limits is vital for evaluating the integral correctly and obtaining the intended result.
Initially, we are integrating from 0 to 1 in terms of \( x \). After substituting \( u = 2x + 1 \), we need to recalculate these limits for \( u \). Using the substitution variable, when \( x = 0 \), \( u = 2(0) + 1 = 1 \), and when \( x = 1 \), \( u = 2(1) + 1 = 3 \).
This change means that our rewritten integral now spans from \( u = 1 \) to \( u = 3 \), accurately reflecting the original integral's scope but in terms of \( u \) instead of \( x \). Ensuring correct limits is vital for evaluating the integral correctly and obtaining the intended result.
Integrand Simplification
Integrand simplification is a strategic method to make integration straightforward by transforming a complex expression into a simpler one. This simplification can involve algebraic manipulation, substitution, and factoring.
In the original problem, the integrand \( \frac{1}{\sqrt{2x+1}} \) looks difficult to handle directly. Through substitution, it transforms into \( \frac{1}{\sqrt{u}} \), a simpler form that is easier to integrate.
This simplification reduces potential errors that might arise from manually dealing with the more complex original expression. Therefore, integrand simplification is a key strategy in integration, aimed at making the calculation less prone to mistakes and more approachable.
In the original problem, the integrand \( \frac{1}{\sqrt{2x+1}} \) looks difficult to handle directly. Through substitution, it transforms into \( \frac{1}{\sqrt{u}} \), a simpler form that is easier to integrate.
This simplification reduces potential errors that might arise from manually dealing with the more complex original expression. Therefore, integrand simplification is a key strategy in integration, aimed at making the calculation less prone to mistakes and more approachable.
Other exercises in this chapter
Problem 6
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