Integration of Secant Cubed X

In this tutorial we shall now discuss integral of secant cubed of X, and this integration can be evaluated by using integration by parts. But first break the power of function from cube into square and one power of the function then we can use integration by parts of that function, because we know that formula for integration secant square of X from previous tutorials.

The integral of secant cubed of X is of the form

I = \int {{{\sec }^3}xdx} \,\,\,\,{\text{ - - - }}\left( {\text{i}} \right)

First break the power of function into \sec x and {\sec ^2}x, now the integral (i) becomes

I = \int {\sec x{{\sec }^2}xdx} \,\,\,\,{\text{ - - - }}\left( {{\text{ii}}} \right)

Considering \sec x and {\sec ^2}x as first and second functions respectively from integration by parts, we have

Using formula for integration by parts, we have

\int {\left[ {f\left( x \right)g\left( x \right)} \right]dx = f\left( x \right)\int {g\left( x \right)dx - \int {\left[ {\frac{d}{{dx}}f\left( x \right)\int {g\left( x \right)dx} } \right]dx} } }

Equation (ii) becomes using above formula, we have

\begin{gathered} I = \sec x\int {{{\sec }^2}xdx - \int {\left[ {\frac{d}{{dx}}\sec x\left( {\int {{{\sec }^2}xdx} } \right)} \right]} dx} \\ \Rightarrow I = \sec x\tan x - \int {\left[ {\left( {\sec x\tan x} \right)\tan x} \right]} dx \\ \Rightarrow I = \sec x\tan x - \int {\left[ {\sec x{{\tan }^2}x} \right]} dx \\ \end{gathered}

Now using trigonometric identity {\tan ^2}x = {\sec ^2}x - 1, we have

\begin{gathered} I = \sec x\tan x - \int {\left[ {\sec x\left( {{{\sec }^2}x - 1} \right)} \right]} dx \\ \Rightarrow I = \sec x\tan x - \int {\left[ {{{\sec }^3}x - \sec x} \right]} dx \\ \Rightarrow I = \sec x\tan x - \int {{{\sec }^3}xdx + \int {\sec xdx} } \\ \end{gathered}

But from our original problem I = \int {{{\sec }^3}xdx} , putting this value, we have

\begin{gathered} I = \sec x\tan x - I + \ln \left( {\sec x + \tan x} \right) + c \\ \Rightarrow I + I = \sec x\tan x + \ln \left( {\sec x + \tan x} \right) + c \\ \Rightarrow I = \frac{1}{2}\left[ {\sec x\tan x + \ln \left( {\sec x + \tan x} \right)} \right] + c \\ \Rightarrow \int {{{\sec }^3}xdx} = \frac{1}{2}\left[ {\sec x\tan x + \ln \left( {\sec x + \tan x} \right)} \right] + c \\ \end{gathered}