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Question: What is the integral of \(\int{{{\tan }^{4}}\left( x \right){{\sec }^{2}}\left( x \right)dx}\)?...

What is the integral of ∫tan⁡4(x)sec⁡2(x)dx\int{{{\tan }^{4}}\left( x \right){{\sec }^{2}}\left( x \right)dx}?

Explanation

Solution

Assume the given integral as ‘I’. Now, substitute tan⁡x=k\tan x=k and differentiate both the sides to find the value of sec⁡2xdx{{\sec }^{2}}xdx in terms of dkdk and substitute in the integral. Use the formula d(tan⁡x)dx=sec⁡2x\dfrac{d\left( \tan x \right)}{dx}={{\sec }^{2}}x. Use the basic formula of the integral given as ∫kndk=kn+1n+1\int{{{k}^{n}}dk}=\dfrac{{{k}^{n+1}}}{n+1} to get the answer. Substitute back the assumed value of k to get the function in terms of x. Finally, add the constant of indefinite integration (c) at last.

Complete step by step answer:
Here we have been provided with the function tan⁡4(x)sec⁡2(x){{\tan }^{4}}\left( x \right){{\sec }^{2}}\left( x \right) and we are asked to integrate it. Let us assume the integral as I, so we have,
⇒I=∫tan⁡4(x)sec⁡2(x)dx\Rightarrow I=\int{{{\tan }^{4}}\left( x \right){{\sec }^{2}}\left( x \right)dx}
Let us use the substitution method to solve this integral, so substituting tan⁡x=k\tan x=k and differentiating both the sides to find the value of sec⁡2xdx{{\sec }^{2}}xdx in terms of dkdk we get,
⇒d(tan⁡x)=dk ⇒sec⁡2xdx=dk \begin{aligned} & \Rightarrow d\left( \tan x \right)=dk \\\ & \Rightarrow {{\sec }^{2}}xdx=dk \\\ \end{aligned}
Substituting the assumed and above obtained relation in the integral I we get,
⇒I=∫k4dk\Rightarrow I=\int{{{k}^{4}}dk}
Using the formula ∫kndk=kn+1n+1\int{{{k}^{n}}dk}=\dfrac{{{k}^{n+1}}}{n+1} where n must not be equal to -1, we get,

& \Rightarrow I=\dfrac{{{k}^{4+1}}}{4+1} \\\ & \Rightarrow I=\dfrac{{{k}^{5}}}{5} \\\ \end{aligned}$$ Substituting back the value of k we get, $$\Rightarrow I=\dfrac{{{\tan }^{5}}x}{5}$$ Now, since the given integral is an indefinite integral and therefore we need to add a constant of integration (c) in the expression obtained for I. So we get, $$\Rightarrow I=\dfrac{{{\tan }^{5}}x}{5}+c$$ Hence, the above relation is our answer. **Note:** Note that it will be difficult to find the required integral without the help of a substitution method. This method is generally used when we get a hint while observing the integral that we have the function and its derivative term inside the integral sign. Remember the derivative formula of all the basic functions like: trigonometric and inverse trigonometric functions, exponential functions, log functions etc.