Question

integral of sec Square X by 2​

Answer 1

$\int sec^{2}\dfrac{x}{2}\:dx=2\:tan\dfrac{x}{2}+c$

where $c$ is integration constant.

Step-by-step explanation:

Now, $\int sec^{2}\dfrac{x}{2}\:dx$

• Let, $\dfrac{x}{2}=z$

• Then, $d(\dfrac{x}{2})=d(z)$

• $\Rightarrow \dfrac{1}{2}\:dx=dz$

• $\Rightarrow dx=2\:dz$

$=\int sec^{2}z\:(2\:dz)$

$=2\int sec^{2}z\:dz$

$=2\:tanz+c$, where $c$ is integration constant

• since $\int sec^{2}x\:dx=tanx+c$, where $c$ is integration constant

$=2\:tan\dfrac{x}{2}+c$

• since $z=\dfrac{x}{2}$

$\Rightarrow \boxed{\int sec^{2}\dfrac{x}{2}\:dx=2\:tan\dfrac{x}{2}+c}$

This is the required integral.

NOTE:

We can apply the following formula to solve the problem in a go.

$\quad\boxed{\int sec^{2}(mx)\:dx=\dfrac{1}{m}tan(mx)+c}$, where $c$ is constant of integration.