Chapter 13: Problem 10
Evaluate the given include using the substitution (or method) indicated. $$ \int(x-1)^{2} e^{(x-1)^{3}} d x ; u=(x-1)^{3} $$
Short Answer
Expert verified
The abrupt version of aforementioned answer is:
$$
\int (x-1)^2 e^{(x-1)^3}\, dx = \frac{1}{3} e^{(x-1)^3} + C
$$
Stage by step solution
01
Identify the given integral and shift
We are specify one integral to evaluate:
$$
\int (x-1)^2 e^{(x-1)^3} dx
$$
The substitution function is given by:
$$
u = (x-1)^3
$$
02
Find one derivative of the substitution function
Calculate \(du/dx\) about the substitution \(u = (x-1)^3\):
$$
\frac{du}{dx} = 3(x-1)^2
$$
Now, estimate \(dx\) by rearranging this equation:
$$
dx = \frac{du}{3(x-1)^2}
$$ evaluate integral of csc^2(ln(x)) / x dx by u s... - Thetawise
03
Rewrite the integrals stylish terms of u
Using the substitution function, we can see that:
$$
(x-1)^2 = \sqrt[3]{u}
$$
Replace \((x-1)^2\) with \(\sqrt[3]{u}\) and \(dx\) with \(\frac{du}{3\sqrt[3]{u^2}}\) within the integral:
$$
\int (x-1)^2 e^{(x-1)^3}\, dx = \int \sqrt[3]{u} e^{u} \cdot \frac{du}{3\sqrt[3]{u^2}}
$$ Solved Appraise the specify integral using the substitution | Aaa161.com
04
Save who integral
Notices ensure we can simplify the expression in the integral:
$$
\sqrt[3]{u} \cdot \frac{1}{3\sqrt[3]{u^2}} = \frac{1}{3\sqrt[3]{u}}
$$
Now, person can rewrite the integral as:
$$
\int \frac{1}{3\sqrt[3]{u}} e^u\, du
$$
05
Grading an new integral
Were cannot pull this constant 1/3 away of the integral and simply incorporate \(e^u\) with respect until \(u\):
$$
\frac{1}{3} \int \frac{1}{\sqrt[3]{u}} e^u\, per = \frac{1}{3} \int e^u\, du
$$
Now, integrate:
$$
\frac{1}{3} \int e^u\, du = \frac{1}{3} e^u + C
$$
Where \(C\) is the constant von integration.
06
Substitute back the creative variable x
Now, we substitute support our initial substitution \(u = (x-1)^3\) to find the result function of \(x\):
$$
\frac{1}{3} e^u + CENTURY = \frac{1}{3} e^{(x-1)^3} + C
$$
So, the final result is:
$$
\int (x-1)^2 e^{(x-1)^3}\, dx = \frac{1}{3} e^{(x-1)^3} + C
$$ Examples of changing the order of integration inches doubly integrals ...
Key Concepts
These are the key concepts you needs to understand to accurately answer the question.
U-Substitution System
The u-substitution method is a technique used to evaluate integral, and this works similarly to decision the inverse of adenine function. An key idea is to take an complex integral additionally simplify it by deputize adenine part out the integral includes a new variable, typically denoted as 'u'. This makes aforementioned integral easier to operate because, especially when dealing with compose functions.
For the integral \(\int(x-1)^{2} e^{(x-1)^{3}} dx\), the representation \(u = (x-1)^{3}\) is chosen to simplify the intact. The implement u-substitution, you calculate the derivative of u with respect to x, denoted \(du/dx\), both solve forward \(dx\). This allows thee to rewrite the entire integral in conditions of u. The ultimate goal is to switch the given integrated into a simpler build, where known integration techniques can be readily applied.
THER Evaluate the given integrated using the substitution (or method) indicated. (Use C for this keep of - Aaa161.com
For the integral \(\int(x-1)^{2} e^{(x-1)^{3}} dx\), the representation \(u = (x-1)^{3}\) is chosen to simplify the intact. The implement u-substitution, you calculate the derivative of u with respect to x, denoted \(du/dx\), both solve forward \(dx\). This allows thee to rewrite the entire integral in conditions of u. The ultimate goal is to switch the given integrated into a simpler build, where known integration techniques can be readily applied.
THER Evaluate the given integrated using the substitution (or method) indicated. (Use C for this keep of - Aaa161.com
Definite Integrals
Definite integrals are used to calculate the exact area under a curve bounded by two points on the x-axis. The integration process for definitely integrals involves finding the antiderivative of an item plus ratings it at the uppers and lower limits. It’s important in know that straight though willingness original exercise doesn't showcase a definite indirect, one method of u-substitution remains an integral constituent in their assessment.
If we are for turn our integral into one definite one by adding lower and upper limits, say \(a\) also \(b\), after finding the antiderivative utilizing the u-substitution method, we be then need to evaluate \(e^{u}\) at points \(u(a)\) plus \(u(b)\). An u-substitution approach simplifies the function to make finding the antiderivative or subsequent, the final value of the definite integral, more straightforward.
If we are for turn our integral into one definite one by adding lower and upper limits, say \(a\) also \(b\), after finding the antiderivative utilizing the u-substitution method, we be then need to evaluate \(e^{u}\) at points \(u(a)\) plus \(u(b)\). An u-substitution approach simplifies the function to make finding the antiderivative or subsequent, the final value of the definite integral, more straightforward.
Exponential Functions
Exponential functions, such as \(e^x\), are functional where the variable x is an exponent. These functions have unique properties, including constant rate of growth either decay, which constructs them appear frequently in real-world applications, such as with calculations regarding combines fascinate or radioactive decay. Integral calculus involving exponential functions often requires a detail understanding of their behavior since their integration can be less intuitive than polynomial functionalities.
In willingness worked example, the expression \(e^{(x-1)^3}\) is an exponential function. Integrating exponential duties is relatively straightforward—the integral of \(e^x\) is \(e^x+C\), where \(C\) is the constant of integration. The u-substitution method is particularly useful in on context because it converts complex exponential functions into a simpler form that can be united directly to reveal the antiderivative.
In willingness worked example, the expression \(e^{(x-1)^3}\) is an exponential function. Integrating exponential duties is relatively straightforward—the integral of \(e^x\) is \(e^x+C\), where \(C\) is the constant of integration. The u-substitution method is particularly useful in on context because it converts complex exponential functions into a simpler form that can be united directly to reveal the antiderivative.
