# The integral integral tan x sec^3 x dx is equal to a. fraction sec^3 x 3 + C b. fraction sec^4...

## Question:

The integral {eq}\displaystyle \int \tan x \sec^3 x \ dx {/eq} is equal to

{eq}\text{a.} \ \dfrac{ \sec^3 x}{3} + \text{C} \\ \text{b.} \ \dfrac{ \sec^4 x}{4} + \text{C} \\ \text{c.} \ \dfrac{1}{2} (\sec x \tan x)^2 + \text{C} \\ \text{d.} \ \dfrac{1}{2} \tan^2 x + \text{C} {/eq}

## Power Rule:

The power rule of integration is an integration technique that is used on terms that are of the form {eq}\displaystyle u^n {/eq}, where u is a variable and n is a constant exponent. The power rule of integration tells us that the integral of this term is:

{eq}\displaystyle \int\ u^n\ du = \frac{u^{n+1}}{n+1} + C {/eq}

Given:

{eq}\displaystyle \int\ \tan\ x\sec^3\ x\ dx {/eq}

We can rewrite this as:

{eq}\displaystyle \int\ \tan\ x\sec^3\ x\ dx = \int\ \sec^2\ x (\sec\ x\tan\ x)\ dx {/eq}

Let:

• {eq}\displaystyle u = sec\ x {/eq}
• {eq}\displaystyle du = \sec\ x\tan\ x\ dx {/eq}

So, we rewrite the integral in terms of u:

{eq}\displaystyle \int\ \tan\ x\sec^3\ x\ dx = \int\ u^2\ du {/eq}

We use the power rule of integration:

{eq}\displaystyle \int\ u^n\ du = \frac{u^{n+1}}{n+1} + C {/eq}

So,

{eq}\displaystyle \int\ \tan\ x\sec^3\ x\ dx = \frac{1}{3} u^3 + C {/eq}

We replace u with the original x variable by reversing the substitution:

{eq}\displaystyle \boxed{a.\ \int\ \tan\ x\sec^3\ x\ dx = \frac{1}{3} \sec^3\ x + C} {/eq}