Integrate Using u-Substitution integral from 1 to 2 of (e^(1/(x^4)))/(x^5) with respect to x
The problem is asking to evaluate a definite integral of the function (e^(1/(x^4)))/(x^5) with respect to x, between the limits of x = 1 and x = 2, using the method of u-substitution. U-substitution is a technique used in calculus to simplify integrals by substituting the complicated part of the function with a new variable 'u', so that the integral becomes easier to solve. This typically involves identifying a part of the integrand (the function being integrated) that can be differentiated to find 'du', which is another part of the integrand. Once 'u' and 'du' are identified, the original integral is rewritten in terms of 'u', which should be easier to integrate. After integrating with respect to 'u', one must substitute back the original variables to find the definite integral with the original limits of integration.
$\int_{1}^{2} \frac{e^{\frac{1}{x^{4}}}}{x^{5}} d x$
Answer
Solution:
Step 1
Choose $u = \frac{1}{x^{4}}$. Consequently, we have $d u = - \frac{4}{x^{5}} d x$, which implies $d x = - \frac{1}{4} x^{5} d u$. We will now express the integral in terms of $u$ and $d u$.
Step 1.1
Define $u$ as $u = \frac{1}{x^{4}}$ and compute $\frac{d u}{d x}$.
Step 1.1.1
Calculate the derivative of $\frac{1}{x^{4}}$ as $\frac{d}{d x} \left( \frac{1}{x^{4}} \right)$.
Step 1.1.2
Utilize the rules for manipulating exponents.
Step 1.1.2.1
Express $\frac{1}{x^{4}}$ in the form $(x^{4})^{-1}$ and differentiate: $\frac{d}{d x} \left( (x^{4})^{-1} \right)$.
Step 1.1.2.2
Multiply the exponent in $(x^{4})^{-1}$ accordingly.
Step 1.1.2.2.1
Invoke the power rule for exponents, where $(a^{m})^{n} = a^{mn}$, and apply it: $\frac{d}{d x} (x^{-4})$.
Step 1.1.2.2.2
Multiply the exponent $4$ by $-1$: $\frac{d}{d x} (x^{-4})$.
Step 1.1.3
Differentiate using the Power Rule, which states $\frac{d}{d x} (x^{n}) = n x^{n - 1}$ for $n = -4$: $-4 x^{-5}$.
Step 1.1.4
Simplify the expression.
Step 1.1.4.1
Rewrite using the rule for negative exponents, $b^{-n} = \frac{1}{b^{n}}$: $-4 \frac{1}{x^{5}}$.
Step 1.1.4.2
Combine the terms.
Step 1.1.4.2.1
Merge $-4$ with $\frac{1}{x^{5}}$: $-\frac{4}{x^{5}}$.
Step 1.1.4.2.2
Place the negative sign in front of the fraction: $-\frac{4}{x^{5}}$.
Step 1.2
Insert the lower limit into $u = \frac{1}{x^{4}}$: $u_{\text{lower}} = \frac{1}{1^{4}}$.
Step 1.3
Perform simplification.
Step 1.3.1
Any number raised to any power is itself: $u_{\text{lower}} = \frac{1}{1}$.
Step 1.3.2
Divide $1$ by $1$: $u_{\text{lower}} = 1$.
Step 1.4
Insert the upper limit into $u = \frac{1}{x^{4}}$: $u_{\text{upper}} = \frac{1}{2^{4}}$.
Step 1.5
Compute $2$ to the fourth power: $u_{\text{upper}} = \frac{1}{16}$.
Step 1.6
Use the values of $u_{\text{lower}}$ and $u_{\text{upper}}$ to evaluate the definite integral: $u_{\text{lower}} = 1$, $u_{\text{upper}} = \frac{1}{16}$.
Step 1.7
Reformulate the integral with $u$, $d u$, and the new integration bounds: $\int_{1}^{\frac{1}{16}} - \frac{1}{4} e^{u} d u$.
Step 2
Extract the constant $- \frac{1}{4}$ from the integral: $- \frac{1}{4} \int_{1}^{\frac{1}{16}} e^{u} d u$.
Step 3
Integrate $e^{u}$ with respect to $u$: $- \frac{1}{4} e^{u} \bigg|_{1}^{\frac{1}{16}}$.
Step 4
Evaluate $e^{u}$ at the bounds $\frac{1}{16}$ and $1$: $- \frac{1}{4} \left( e^{\frac{1}{16}} - e^{1} \right)$.
Step 5
Simplify the result: $- \frac{1}{4} \left( e^{\frac{1}{16}} - e \right)$.
Step 6
Present the final answer in various formats.
Exact Form: $- \frac{1}{4} \cdot \left( e^{\frac{1}{16}} - e \right)$ Decimal Form: $0.41344684 \ldots$
Knowledge Notes:
To solve the given integral using u-substitution, we follow these steps:
Choose a substitution: Identify a part of the integral that can be substituted with a variable $u$ to simplify the integral. This often involves choosing a function inside a composition that, when differentiated, appears elsewhere in the integral.
Differentiate $u$: Find the derivative of $u$ with respect to $x$, denoted as $\frac{d u}{d x}$, to replace $dx$ in the integral.
Rewrite the integral: Express the integral entirely in terms of $u$ and $du$. This may involve changing the limits of integration if the integral is definite.
Integrate with respect to $u$: Perform the integration now that the integral is in terms of $u$.
Back-substitute: If the integral was indefinite, substitute back the original $x$-related expression for $u$. If the integral was definite, evaluate the antiderivative at the new limits of integration.
Simplify: Simplify the expression to get the final result.
In this problem, the substitution $u = \frac{1}{x^{4}}$ simplifies the integral because the derivative of $u$ is proportional to another part of the integrand, allowing us to replace the complex $x$-dependent expression with a simpler $u$-dependent one. After performing the integration, we substitute the limits of integration into the antiderivative and simplify to find the exact or decimal form of the result.
