Find the integral of the function tan4 x - Vidyadip

Question

Find the integral of the function tan⁴ x

✓

Answer

tan⁴x = tan²x.tan²x
= (sec²x - 1) tan²x
= sec²x tan²x - tan²x
= sec²x tan²x - (sec²x - 1)
= sec²x tan²x - sec²x + 1
Now, $\int \tan ^{4} x d x=\int \sec ^{2} x \tan ^{2} x d x-\int \sec ^{2} x d x+\int 1 d x$
$= \int \sec ^{2} x \tan ^{2} x d x-\tan x+x+C$
Now, let tanx = t
$\Rightarrow$ sec²x dx = dt
$\Rightarrow \int \sec ^{2} x \tan ^{2} x d x=\int t^{2} d t=\frac{t^{3}}{3}=\frac{\tan ^{3} x}{3}$
$\Rightarrow \int \tan ^{4} x d x=\frac{1}{3} \tan ^{3} x-\tan x+x+C$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks" from Integrals→Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

For each of the differential equations in find the general solution:
(e^x + e^–x) dy – (e^x – e^–x) dx = 0

Test the continuity of the function on f(x) at the origin:
$\text{f}\ (\text{x})=\begin{cases}\frac{\text{x}}{\text{|x|}},& \text{x}\neq0\\1, & \text{x} = 0\end{cases}$

A pair of dice is thrown. Find the probability of getting 7 as the sum if it is known that the second die always exhibits a prime number.

Solve system of linear equations, using matrix method.

Evaluate the following integrals:
$\int_{0}^\limits{1}\frac{\sqrt{\tan^{-1}\text{x}}}{1+\text{x}^2}\text{ dx}$

Write the value of $\sin^{-1}\Big(\frac{-\sqrt3}{2}\Big)+\cos^{-1}\Big(\frac{-1}{2}\Big)$

If $\vec{\text{a}}=2\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}},\vec{\text{b}}=-\hat{\text{i}}+2\hat{\text{j}}+\hat{\text{k}},$and $\vec{\text{c}}=3\hat{\text{i}}+\hat{\text{j}}$ are such that $\vec{\text{a}}+\lambda\vec{\text{b}}$ is perpendicular to $\vec{\text{c}},$ then find the value of $\lambda.$

Let $\text{A}=\begin{bmatrix}2&-3\\-7&5\end{bmatrix}$ and $\text{B}=\begin{bmatrix}1&0\\2&-4\end{bmatrix},$ verify that
$(\text{A}-\text{B})^\text{T}=\text{A}^\text{T}-\text{B}^\text{T}$

Assume that on an average one telephone number out of 15 called between 2 P.M. and 3 P.M. on week days is busy. What is the probability that if six randomly selected telephone numbers are called, at least 3 of them will be busy?

Find the area enclosed by the circle x² + y² = a²