Problem 56

Question

Use substitution to evaluate the definite integrals. $$ \int_{1}^{2} \frac{x d x}{\left(x^{2}+1\right) \ln \left(x^{2}+1\right)} $$

Step-by-Step Solution

Verified

Answer

$ \frac{1}{2} \ln \left( \frac{\ln(5)}{\ln(2)} \right) $

1Step 1: Identify the Substitution

We notice the composition of functions within the integral. Specifically, let us set the substitution as follows: $ u = x^2 + 1 $.

2Step 2: Differentiate the Substitution

To replace $ dx $ in terms of $ du $, differentiate $ u $ with respect to $ x $ to get: $ du = 2x \, dx $. Therefore, $ x \, dx = \frac{1}{2} \, du $.

3Step 3: Change the Limits of Integration

Convert the limits from $ x $ to $ u $. When $ x = 1 $, $ u = 1^2 + 1 = 2 $. When $ x = 2 $, $ u = 2^2 + 1 = 5 $. Thus, the integral is redefined from $ u = 2 $ to $ u = 5 $.

4Step 4: Substitute and Rewrite the Integral

Substitute $ u $ and $ du $ back into the integral. The integral becomes $ \int_{2}^{5} \frac{1}{2} \cdot \frac{1}{u \ln(u)} \, du $. This simplifies to $ \frac{1}{2} \int_{2}^{5} \frac{1}{u \ln(u)} \, du $.

5Step 5: Integrate the Expression

Recognize that the integral $ \int \frac{1}{u \ln(u)} \, du $ is a standard integral which evaluates to $ \ln|\ln(u)| + C $. Thus, our integral is $ \frac{1}{2} \ln|\ln(u)| \Big|_{2}^{5} $.

6Step 6: Evaluate the Definite Integral

Compute the result by evaluating the expression at the bounds: $ \frac{1}{2} \left( \ln|\ln(5)| - \ln|\ln(2)| \right) $.

7Step 7: Simplify the Expression

Using logarithm properties, this can be rewritten as: $ \frac{1}{2} \ln \left( \frac{\ln(5)}{\ln(2)} \right) $. Complete the simplification to obtain the final result.

Key Concepts

Substitution MethodLimits of IntegrationU-SubstitutionStandard Integrals

Substitution Method

The substitution method is a powerful technique for evaluating integrals, especially when dealing with functions that involve compositions or complicated products. The main idea is to simplify the integral by changing the variable of integration. This is achieved by identifying a part of the integrand that can be represented as a single variable. In our example, we noticed that the expression $x^2 + 1$ appears both in the polynomial and the logarithm. This suggests using $u = x^2 + 1$ as our substitution.
Once you identify your substitution, the next step is to express $dx$ in terms of the new variable $du$. This involves differentiating $u$ with respect to $x$, which gives us $du = 2x \, dx$. Consequently, we solve for $x \, dx$ and find $x \, dx = \frac{1}{2} \, du$. Substituting these values back into the integral makes the process of integration much simpler. The substitution method streamlines your calculations by turning a complicated integral into a more manageable form.

Limits of Integration

When you use substitution with definite integrals, you must also change the limits of integration to match your new variable $u$. This step is crucial and helps ensure that the evaluation is performed correctly. It not only maintains the definition of the definite integral but also simplifies computations.
In our integral, the original limits are $x = 1$ and $x = 2$. By substituting into $u = x^2 + 1$, these limits transform into $u = 2$ when $x = 1$ and $u = 5$ when $x = 2$. Thus, the limits of integration in terms of $u$ are now $2$ to $5$. Remember, if you omit this step, your integral will potentially be incorrect as it won’t account for the change of variables.

U-Substitution

U-substitution, a particular case of the substitution method, is extremely useful for simplifying integrals involving composite functions. You aim to rewrite the integral in terms of a single variable $u$, making complex expressions easier to manage and integrate. This technique requires two main steps:

Identify an expression inside the integral to substitute with $u$, simplifying the expression significantly. For instance, in our integral, $x^2 + 1$ was chosen.
Calculate the differential $du$ in terms of $dx$, allowing you to completely express the integral in terms of $u$. Here, $x \, dx = \frac{1}{2} \, du$.

Once the substitution is applied, the integral becomes easier to evaluate, transforming challenging problems into ones you can handle with standard techniques. Always remember to revert back to the original variable if the problem specifies finding an indefinite integral.

Standard Integrals

Standard integrals are a set of known integral forms that you can use to simplify and directly solve many integration problems without deriving results from scratch. Familiarizing yourself with these can greatly reduce problem-solving time in calculus.
In our case, after applying u-substitution, we arrived at the integral $\int \frac{1}{u \ln(u)} \, du$, a standard form that integrates to $\ln|\ln(u)| + C$. Recognizing this allows us to quickly find a solution without going through lengthy calculations.

The key is identifying when your integral matches a standard form, enabling you to apply known results directly.
After integrating, apply the new limits to evaluate the definite integral from the given lower limit to the upper limit.

Using standard integrals optimizes solving procedures, saving time and effort by leveraging already established results.

Problem 56

Problem 57

Other exercises in this chapter

Problem 55

Use substitution to evaluate the definite integrals. $$ \int_{e}^{e^{2}} \frac{d x}{x(\ln x)^{2}} $$

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Problem 56

Use either substitution or integration by parts to evaluate each integral. $$ \int \frac{1}{x^{2}+5} d x $$

View solution

Problem 57

Use either substitution or integration by parts to evaluate each integral. $$ \int \frac{x}{x+3} d x $$

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Problem 57

Use substitution to evaluate the definite integrals. $$ \int_{1}^{9} \frac{1}{\sqrt{x}} e^{-\sqrt{x}} d x $$

View solution