# Evaluate $2\int {{{\tan }^4}t{{\sec }^2}tdt} $.

Last updated date: 17th Mar 2023

•

Total views: 205.2k

•

Views today: 2.85k

Answer

Verified

205.2k+ views

**Hint:**The given question requires us to integrate a trigonometric function of $t$ with respect to $t$. Integration gives us a family of curves. Integrals in maths are used to find many useful quantities such as areas, volumes, displacement, etc. integral is always found with respect to some variables, which in this case is $x$.

**Complete step by step solution:**

The given question requires us to integrate a trigonometric function ${\tan ^4}t{\sec ^2}t$ in variable t.

Now, the integral given to us is complex and cannot be solved directly using basic results of integration of functions. We will have to simplify the function so as to solve the integral.

So, we can assign a new variable in the integral.

Let us assume $\tan t = x$.

Then, differentiating both sides of the equation, we get,

$ \Rightarrow {\sec ^2}tdt = dx$

Hence, the integral given is,

$2\int {{{\tan }^4}t{{\sec }^2}tdt} $

Substituting the value of ${\sec ^2}tdt$ in terms of x as ${\sec ^2}tdt = dx$ and the value of $\tan t$ as x, we get,

$ \Rightarrow 2\int {{x^4}dx} $

Now, we know that the integral of ${x^n}$ with respect to x is $\dfrac{{{x^{n + 1}}}}{{n + 1}}$ using the power rule of integration. So, we get,

$ \Rightarrow 2\left( {\dfrac{{{x^5}}}{5}} \right) + c$, where c is an arbitrary constant.

Substituting the value of x in the expression as $\tan t$, we get,

$ \Rightarrow 2\left( {\dfrac{{{{\tan }^5}t}}{5}} \right) + c$

Opening the bracket, we get,

$ \Rightarrow \dfrac{{2{{\tan }^5}t}}{5} + c$

So, $\dfrac{{2{{\tan }^5}t}}{5} + c$, where c is any arbitrary constant, is the value of the given integral, $2\int {{{\tan }^4}t{{\sec }^2}tdt} $.

**Note:**

The indefinite integrals of certain functions may have more than one answer in different forms. However, all these forms are correct and interchangeable into one another. Indefinite integral gives us the family of curves as we don’t know the exact value of the constant.

Recently Updated Pages

If ab and c are unit vectors then left ab2 right+bc2+ca2 class 12 maths JEE_Main

A rod AB of length 4 units moves horizontally when class 11 maths JEE_Main

Evaluate the value of intlimits0pi cos 3xdx A 0 B 1 class 12 maths JEE_Main

Which of the following is correct 1 nleft S cup T right class 10 maths JEE_Main

What is the area of the triangle with vertices Aleft class 11 maths JEE_Main

KCN reacts readily to give a cyanide with A Ethyl alcohol class 12 chemistry JEE_Main

Trending doubts

What was the capital of Kanishka A Mathura B Purushapura class 7 social studies CBSE

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Tropic of Cancer passes through how many states? Name them.

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

Name the Largest and the Smallest Cell in the Human Body ?