Problem 8 Evaluate the indefinite integral... [FREE SOLUTION]

Chapter 6: Problem 8

Evaluate the indefinite integral to develop an understanding of Substitution. $$ \int \frac{1}{\sqrt{2 x+3}} d x $$

Short Answer

Expert verified

The integral is $(2x + 3)^{\frac{1}{2}} + C$.

Step by step solution

Identify the function to be substituted

Consider the expression under the square root, which is $2x + 3$. We will set this equal to a new variable: $ u = 2x + 3 $. This will simplify the integral.

Differentiate the substitution expression

Differentiate the expression $ u = 2x + 3 $ with respect to $ x $ to find the differential $ du $ in terms of $ dx $. This gives us $ \frac{du}{dx} = 2 $, hence $ du = 2dx $.

Solve for dx

Rearrange the expression $ du = 2 dx $ to solve for $ dx $. Thus, $ dx = \frac{1}{2}du $.

Substitute into the integral

Substitute $ u $ and $ dx $ into the original integral. The original integral $ \int \frac{1}{\sqrt{2x+3}} \, dx $ becomes $ \int \frac{1}{\sqrt{u}} \cdot \frac{1}{2} \, du $.

Simplify and integrate

Factor out the constant $ \frac{1}{2} $: $ \frac{1}{2} \int u^{-\frac{1}{2}} \, du $. Integrate $ u^{-\frac{1}{2}} $ to get $ 2u^{\frac{1}{2}} + C $. Combined with the constant factor, it becomes $ u^{\frac{1}{2}} + C $.

Back-substitute for the original variable

Replace $ u $ with $ 2x + 3 $, the original substitution: $ (2x + 3)^{\frac{1}{2}} + C $. This represents the indefinite integral in terms of $ x $.

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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 technique used in calculus to simplify the process of evaluating integrals. By transforming the integral into a simpler form, it becomes easier to handle. In our problem, the expression inside the square root, $2x + 3$, is replaced with a new variable, $u$.

Identify the complex part of the integral: the expression $2x + 3$ in $\int \frac{1}{\sqrt{2x+3}} \, dx$.
Set $u = 2x + 3$, transforming the integrand into a simpler expression.

This substitution helps us focus on integrating a function of $u$, making the whole process much more manageable.

Differentiation

Differentiation is key in the substitution method, as it allows us to relate the variables $x$ and $u$. By differentiating the substitution expression $u = 2x + 3$ with respect to $x$, we find the relationship between $dx$ and $du$. The derivative $\frac{du}{dx} = 2$ gives us $du = 2dx$. This relationship is crucial for replacing $dx$ and ensuring the integral remains valid, letting us rewrite $dx$ as $\frac{1}{2} du$. This step ensures a clear path for substituting every term.

Integration

Once the substitution and differentiation are done, the integral takes on a simpler form: $\int \frac{1}{\sqrt{u}} \cdot \frac{1}{2} \, du$. Factor out constants to simplify further:

Recognize $\int u^{-1/2} \, du$ is a basic power rule problem.
Apply the power rule: the integral of $u^{-1/2}$ is $2u^{1/2} + C$.

Combining these with the constant factor $\frac{1}{2}$, we obtain $u^{1/2} + C$, where $C$ is the constant of integration.

Variable Substitution

After integrating in terms of $u$, it's time to reintroduce the original variable $x$. This step is known as back-substitution.Replace $u$ with the original expression $2x + 3$ that was substituted, resulting in $(2x + 3)^{1/2} + C$. This final expression represents the indefinite integral in terms of $x$, concluding the substitution process.

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91影视

Evaluate the indefinite integral to develop an understanding of Substitution. $$ \int \frac{1}{\sqrt{2 x+3}} d x $$

Short Answer

Step by step solution

Identify the function to be substituted

Differentiate the substitution expression

Solve for dx

Substitute into the integral

Simplify and integrate

Back-substitute for the original variable

Key Concepts

Substitution Method

Differentiation

Integration

Variable Substitution

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