Question

The integral $\int \frac{{\sin }^{2}x}{{\cos }^{4}x}dx$ is a?

• A. Polynomial of degree $5$ in $\sin x$
• B. Polynomial of degree $4$ in $\tan x$
• C. Polynomial of degree $3$ in $\tan x$
• D. Polynomial of degree $4$ in $\cos x$

Answer 1

The correct option is C Polynomial of degree $3$ in $\tan x$

The Integral $\int \frac{{\sin }^{2}x}{{\cos }^{4}x}dx$ is a ?

We are given $I=\int \frac{{\sin }^{2}x}{{\cos }^{4}x}dx$

as we know $\frac{\sin \theta }{\cos \theta }=\tan \theta$ & $\frac{1}{\cos \theta }=\sec \theta$

$\therefore I=\int {\tan }^{2}x.{\sec }^{2}xdx$

$=\int (\tan x{)}^2.{\sec }^{2}xdx$

now, let $f\left(x\right)=\tan x$

then its differentiation

wrt, $x$, we get,

$f\left(x\right)={\sec }^{2}x$

$I=\int \left[F\right(x){]}^2.F(x)dx$

So, we know its result that $I=\frac{\left[F\right(x){]}^{n+1}}{n+1}+c$

$I=\frac{\left[F\right(x){]}^{2+2}}{2+1}+c$

$I=\frac{\left[F\right(x){]}^3}{3}+c$

$I=\frac{(\tan x{)}^3}{3}+c$

So, The Integral $I$ is a polynomial of degree $3$ in $\tan x$