Evaluate ∫sec(2x) dx

Use Substitution to Simplify the Integral

In this case, the argument of the secant function is $2x$, not just $x$. Because of this constant multiple, the standard integral formula for the secant function cannot be applied directly. The integral can be evaluated by simplifying the function using the substitution method.

Let $u = 2x$

Now, differentiate both sides with respect to $x$ to express $dx$ in terms of $du$ and convert the integral from the variable $x$ to $u$.

$\implies$ $\dfrac{d}{dx}{(u)} \,=\, \dfrac{d}{dx}{(2x)}$

$\implies$ $\dfrac{du}{dx} \,=\, \dfrac{d}{dx}{(2 \times x)}$

The variable $x$ is multiplied by the constant $2$. Therefore, when differentiating with respect to $x$, this constant multiple must be taken into account. Using the constant multiple rule of differentiation, the constant can be separated from the differentiation.

$\implies$ $\dfrac{du}{dx} \,=\, 2 \times \dfrac{d}{dx}{(x)}$

$\implies$ $\dfrac{du}{dx} \,=\, 2 \times \dfrac{dx}{dx}$

$\implies$ $\dfrac{du}{dx} \,=\, 2 \times 1$

$\implies$ $\dfrac{du}{dx} \,=\, 2$

Now, we determine the differential element $dx$ in terms of $u$ by applying basic algebraic operations such as multiplication and division.

$\implies$ $\dfrac{du}{dx} \times dx \,=\, 2 \times dx$

$\implies$ $du \times \dfrac{dx}{dx} \,=\, 2 \times dx$

$\implies$ $du \times 1 \,=\, 2 \times dx$

$\implies$ $du \,=\, 2 \times dx$

$\implies$ $\dfrac{du}{2} \,=\, \dfrac{2 \times dx}{2}$

$\implies$ $\dfrac{du}{2} \,=\, \dfrac{\cancel{2} \times dx}{\cancel{2}}$

$\implies$ $\dfrac{du}{2} \,=\, dx$

$\,\,\,\,\therefore\,\,\,\,$ $dx \,=\, \dfrac{du}{2}$

Since the expressions for $2x$ and $dx$ are now known in terms of $u$, we substitute them into the integral to simplify the function for integration.

$\implies$ $\displaystyle \int{\sec{(2x)}}\,dx$ $\,=\,$ $\displaystyle \int{\sec{(u)}}\,\dfrac{du}{2}$

Integrate the Secant Function Using the Standard Formula

The function expressed in terms of $u$ is not yet in a standard form for integration. Therefore, let us first simplify the function further before proceeding with the integration. Because $2$ is a constant multiplier in the integrand, it can be taken outside the integral, which reduces the expression to a standard integrable form.

$\implies$ $\displaystyle \int{\sec{(u)}}\,\dfrac{du}{2}$ $\,=\,$ $\displaystyle \int{\sec{(u)}}\,\dfrac{1 \times du}{2 \times 1}$

Now, we apply the inverse operation of multiplication to rewrite the fraction as a product of two fractions.

$=\,\,$ $\displaystyle \int{\sec{(u)}} \times \dfrac{1}{2} \times \dfrac{du}{1}$

$=\,\,$ $\displaystyle \int{\sec{(u)}} \times \dfrac{1}{2} \times du$

The integral contains a product of three terms: the constant $1/2$, the function $\sec{(u)}$, and the differential $du$. Using the commutative property of multiplication, we can rearrange the order of these terms to simplify the integral.

$=\,\,$ $\displaystyle \int{\dfrac{1}{2} \times \sec{(u)}} \times du$

$=\,\,$ $\displaystyle \int{\dfrac{1}{2} \times \sec{(u)}}\,du$

$=\,\,$ $\displaystyle \dfrac{1}{2} \times \int{\sec{(u)}}\,du$

The fraction $1$ over $2$ is part of the integrand, and it can be factored out of the integral using the constant multiple rule of integration.

$=\,\,$ $\displaystyle \dfrac{1}{2}\int{\sec{u}}\,du$

Finally, apply the standard formula for the integral of the secant function to find the integral of the secant of $u$ with respect to $u$.

$=\,\,$ $\dfrac{1}{2}\ln{|\sec{u}+\tan{u}|}$ $+$ $c$

Back-Substitute and Find the Final Integral of the Given Function

The integral has now been evaluated in terms of $u$, but the original function is in terms of $x$. Therefore, we substitute $u = 2x$ to express the integral completely in terms of $x$.

$=\,\,$ $\dfrac{1}{2}\ln{|\sec{(2x)}+\tan{(2x)}|}$ $+$ $c$