Integrate Using u-Substitution integral from 1 to 2 of (e^(1/(x^4)))/(x^5) with respect to x

Solution:

Step 1

Choose $u=\frac{1}{x^4}$. Consequently, we have $du=-\frac{4}{x^5}dx$, which implies $dx=-\frac{1}{4}{x}^{5}du$. We will now express the integral in terms of $u$ and $du$.

Step 1.1

Define $u$ as $u=\frac{1}{x^4}$ and compute $\frac{du}{dx}$.

Step 1.1.1

Calculate the derivative of $\frac{1}{x^4}$ as $\frac{d}{dx}\left(\frac{1}{x^4}\right)$.

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: $\frac{d}{dx}((x^4{)}^{-1})$.

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: $\frac{d}{dx}(x^{-4})$.

Step 1.1.2.2.2

Multiply the exponent $4$ by $-1$: $\frac{d}{dx}(x^{-4})$.

Step 1.1.3

Differentiate using the Power Rule, which states $\frac{d}{dx}(x^n)=n{x}^{n-1}$ for $n=-4$: $-4x^{-5}$.

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 $u$, $du$, and the new integration bounds: ${\int }_1^{\frac{1}{16}}-\frac{1}{4}{e}^{u}du$.

Step 2

Extract the constant $-\frac{1}{4}$ from the integral: $-\frac{1}{4}{\int }_1^{\frac{1}{16}}{e}^{u}du$.

Step 3

Integrate $e^u$ with respect to $u$: $-\frac{1}{4}{e}^u{|}_1^{\frac{1}{16}}$.

Step 4

Evaluate $e^u$ at the bounds $\frac{1}{16}$ and $1$: $-\frac{1}{4}(e^{\frac{1}{16}}-e^1)$.

Step 5

Simplify the result: $-\frac{1}{4}(e^{\frac{1}{16}}-e)$.

Step 6

Present the final answer in various formats.

Exact Form: $-\frac{1}{4}\cdot (e^{\frac{1}{16}}-e)$ Decimal Form: $0.41344684\dots$

Knowledge Notes:

To solve the given integral using u-substitution, we follow these steps:

1. 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.

2. Differentiate $u$: Find the derivative of $u$ with respect to $x$, denoted as $\frac{du}{dx}$, to replace $dx$ in the integral.

3. 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.

4. Integrate with respect to $u$: Perform the integration now that the integral is in terms of $u$.

5. 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.

6. 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.