Integrate Using u-Substitution integral of sec(2x)tan(2x) with respect to x
The given problem is to perform integration of the function sec(2x)tan(2x) with respect to x using the method of u-substitution. This method involves substituting a part of the integrand (the function being integrated) with a new variable u, which simplifies the integral and makes it easier to solve. The problem requires finding a suitable substitution that will transform the original integral into one that is easier to integrate, and then carrying out the integration with respect to this new variable u. After integrating, the solution will then be converted back from u to x to get the final result.
$\int sec \left(\right. 2 x \left.\right) tan \left(\right. 2 x \left.\right) d x$
Answer
Solution:
Step 1
Assign $u = \sec(2x)$. Consequently, $du = 2 \sec(2x)\tan(2x)dx$, which implies $\frac{1}{2}du = \sec(2x)\tan(2x)dx$. Substitute $u$ and $du$ into the integral.
Step 1.1
Set $u = \sec(2x)$ and compute $\frac{du}{dx}$.
Step 1.1.1
Take the derivative of $\sec(2x)$, $\frac{d}{dx}[\sec(2x)]$.
Step 1.1.2
Employ the chain rule, which states that $\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)$, where $f(x) = \sec(x)$ and $g(x) = 2x$.
Step 1.1.2.1
Using the chain rule, let $v = 2x$. Then differentiate $\sec(v)$ with respect to $v$ and $2x$ with respect to $x$, $\frac{d}{dv}[\sec(v)]\frac{d}{dx}[2x]$.
Step 1.1.2.2
The derivative of $\sec(v)$ with respect to $v$ is $\sec(v)\tan(v)$, $\sec(v)\tan(v)\frac{d}{dx}[2x]$.
Step 1.1.2.3
Substitute $v$ back with $2x$, $\sec(2x)\tan(2x)\frac{d}{dx}[2x]$.
Step 1.1.3
Proceed with differentiation.
Step 1.1.3.1
Since $2$ is a constant, the derivative of $2x$ with respect to $x$ is $2\frac{d}{dx}[x]$, $\sec(2x)\tan(2x)(2\frac{d}{dx}[x])$.
Step 1.1.3.2
Apply the power rule, which states that $\frac{d}{dx}[x^n] = nx^{n-1}$ where $n=1$, $\sec(2x)\tan(2x)(2 \cdot 1)$.
Step 1.1.3.3
Simplify the expression.
Step 1.1.3.3.1
Multiply $2$ by $1$, $\sec(2x)\tan(2x) \cdot 2$.
Step 1.1.3.3.2
Rearrange to place $2$ in front of $\sec(2x)\tan(2x)$, $2\sec(2x)\tan(2x)$.
Step 1.2
Express the integral in terms of $u$ and $du$, $\int \frac{1}{2} du$.
Step 2
Apply the constant multiple rule in integration, $\frac{1}{2}u + C$.
Step 3
Substitute $u$ back with $\sec(2x)$, $\frac{1}{2}\sec(2x) + C$.
Knowledge Notes:
The u-substitution method is a technique for evaluating integrals. When a function is composed of a function and its derivative, u-substitution can simplify the integral into a more manageable form. The process involves the following steps:
Choose a substitution that simplifies the integral, typically a function inside another function or a function whose derivative is also present in the integral.
Differentiate the chosen substitution to find $du$.
Replace all instances of the original variable and its differential in the integral with the new substitution variable $u$ and its differential $du$.
Integrate with respect to $u$.
Replace $u$ with the original expression to return to the original variable.
The chain rule is a fundamental differentiation rule in calculus used when differentiating composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
The constant multiple rule in integration states that the integral of a constant times a function is equal to the constant times the integral of the function.
The power rule for differentiation is used to differentiate functions of the form $x^n$, where $n$ is any real number. The rule states that the derivative of $x^n$ is $nx^{n-1}$.
In the context of the given problem, these concepts are used to perform u-substitution for the integral of $\sec(2x)\tan(2x)$ with respect to $x$. The substitution $u = \sec(2x)$ is chosen because its derivative, $2\sec(2x)\tan(2x)$, appears in the integral, allowing for a straightforward substitution and integration.
