Problem 16 Evaluate the following integrals... [FREE SOLUTION]

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Calculus and Its Applications

Larry J. Goldstein, David I. Schneider, Nakhle H. Asmar

$Math Studyset 91影视 Explanations$ Math

13 Edition

Chapter 9: Problem 16

Evaluate the following integrals: $\int x^{5} \ln x d x$

Short Answer

Expert verified

$\int x^{5} \ln x dx = \frac{x^{6}}{6} \ln x - \frac{x^{6}}{36} + C$

Step by step solution

Identify the methods

The given integral is $\[\begin{equation} \int x^{5} \ln x d x \end{equation}\]$. This integral requires integration by parts, which is based on the formula $\[\begin{equation} \int u \ dv = uv - \int v \ du \end{equation}\]$.

Choose u and dv

Let $u = \ln x$ and $dv = x^{5} dx$. Then, differentiate $u$ and integrate $dv$:$du = \frac{1}{x} dx, \quad v = \int x^{5} dx = \frac{x^{6}}{6}$\.

Substitute into the integration by parts formula

Substitute $u, dv, du, \text{and} v$ into the integration by parts formula: $\[\begin{equation} \int x^{5} \ln x \ dx = \frac{x^{6}}{6} \ln x - \int \frac{x^{6}}{6} \frac{1}{x} \ dx \end{equation}\]$.

Simplify the integral

Simplify the expression inside the remaining integral: $\[\begin{equation} \int \frac{x^{6}}{6} \frac{1}{x} \ dx = \int \frac{x^{5}}{6} \ dx = \frac{1}{6} \int x^{5} \ dx \end{equation}\]$.\ Integrating $x^{5}$: $\[\begin{equation} \frac{1}{6} \int x^{5} \ dx = \frac{1}{6} \frac{x^{6}}{6} = \frac{x^{6}}{36}\end{equation}\]$.

Write the final answer

Combine the results from the integration by parts steps: $\[\begin{equation} \int x^{5} \ln x \ dx = \frac{x^{6}}{6} \ln x - \frac{x^{6}}{36} + C \end{equation}\]$, where $C \text{is the constant of integration} $.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

integration by parts

Integration by parts is a powerful technique used to integrate products of functions. It is based on the product rule of differentiation. The formula is: \[ \int u \ dv = uv - \int v \ du \]Here are the steps to use integration by parts:

Choose which part of the integrand to call $u$ and which to call $dv$.
Differentiatiate $u$ to get $du$, and integrate $dv$ to get $v$.
Substitute these into the formula $uv - \int v \ du$.
Integrate the remaining integral.

For the integral $ \int x^{5} \ln x \ dx $, we set $u = \ln x$ and $dv = x^{5} dx$. This choice simplifies the problem as $du$ and $v$ become easier to handle. By following the steps, we achieve the final integration result.

definite and indefinite integrals

Integrals are classified as definite or indefinite:

Definite integrals have specific limits of integration and represent the signed area under the curve of a function between two points.
Indefinite integrals do not have limits and represent a family of functions, including an arbitrary constant $C$.

In the given problem, we are solving an indefinite integral, which means our result will include the constant of integration $C$. The indefinite integral of the problem is $ \int x^{5} \ln x \ dx = \frac{x^{6}}{6} \ln x - \frac{x^{6}}{36} + C $.

logarithmic functions

Logarithmic functions are the inverse of exponential functions. They have the form $ \log_b(x) $ and are commonly used in many mathematical applications.

Natural logarithms ($ \ln x $) use the base $e$, where $e \approx 2.718$.

In integration by parts, logarithmic functions often play the role of $u$ because their differentiation becomes simpler. For example, in the integral $ \int x^{5} \ln x \ dx $, setting $ u = \ln x $ simplifies the differentiation step, resulting in $ du = \frac{1}{x} dx $. Understanding how to manipulate logarithms is essential in solving integrals involving these functions.

differentiation and integration

Differentiation and integration are fundamental concepts in calculus, acting as inverse processes.

Differentiation finds the rate at which a function is changing at any point.
Integration finds the total accumulation of quantities, such as area under a curve.

When performing integration by parts, one must often differentiate one function and integrate another. For example, in $ \int x^{5} \ln x \ dx $, we differentiate $ \ln x $ and integrate $ x^{5} $. This dual process highlights the interplay between these two operations, reinforcing their interconnectedness in solving complex integrals.

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91影视

Evaluate the following integrals: \(\int x^{5} \ln x d x\)

Short Answer

Step by step solution

Identify the methods

Choose u and dv

Substitute into the integration by parts formula

Simplify the integral

Write the final answer

Key Concepts

integration by parts

definite and indefinite integrals

logarithmic functions

differentiation and integration

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