Question

Please solve this ASAP
and please no fake answers
​

Answer 1

$\\\;\underbrace{\underline{\sf{Understanding\;the\;Concept}}}$

Here the concept of Integration by parts has been used. We see we are given an expression to integrate. Firstly we split the expression into two parts. Then we will integrate both parts separately by rule of integration to find the answer.

Let's do it !!

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★ Identity Used :-

$\\\;\displaystyle{\boxed{\sf{\pink{\int\:u\:v\:dx\;=\;\bf{u\int v\:dx\;-\;\int u'\bigg(\int v\:dx\bigg)dx}}}}}$

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★ Solution :-

Given,

$\\\;\displaystyle{\bf{\green{\int\:\sec^{3}x\:dx}}}$

This can be written as,

$\\\;\displaystyle{\bf{\int\:\sec\:x\:\sec^{2}\:x\:dx}}$

Now firstly let's split this.

• u = sec x

• dv = sec² x dxdx

From this,

» du = sec x tan x dx

And,

» v = tan x

Now we already know the identity that,

$\\\;\displaystyle{\sf{\int\:u\:v\:dx\;=\;\bf{u\int v\:dx\;-\;\int u'\bigg(\int v\:dx\bigg)dx}}}$

By applying this identity into the equation, we get

$\\\;\displaystyle{\sf{\rightarrow\;\;\int\:\sec\:x\:\sec^{2}\:x\:dx\;=\;\bf{\sec\:x\;\tan\:x\;-\;\int \sec\:x\;\tan^{2}\:x\:dx}}}$

This can be written as,

$\\\;\displaystyle{\sf{\rightarrow\;\;\int\:\sec^{3}\:x\:dx\;=\;\bf{\sec\:x\;\tan\:x\;-\;\int \sec\:x\;\tan^{2}\:x\:dx}}}$

We know that,

→ tan² x = sec² x - 1

On applying this here, we get

$\\\;\displaystyle{\sf{\rightarrow\;\;\int\:\sec^{3}\:x\:dx\;=\;\bf{\sec\:x\;\tan\:x\;-\;\int \sec\:x\;(\sec^{2}\:x\;-\;1)\:dx}}}$

$\\\;\displaystyle{\sf{\rightarrow\;\;\int\:\sec^{3}\:x\:dx\;=\;\bf{\sec\:x\;\tan\:x\;-\;\int\:(\sec^{3}\:x\;-\;\sec\:x)\:dx}}}$

$\\\;\displaystyle{\sf{\rightarrow\;\;\int\:\sec^{3}\:x\:dx\;=\;\bf{\sec\:x\;\tan\:x\;-\;\int\:\sec^{3}\:x\:dx\;-\;\sec\:x\:dx}}}$

Now we can integrate as,

$\\\;\displaystyle{\sf{\rightarrow\;\;\int\:\sec^{3}\:x\:dx\;=\;\bf{\sec\:x\;\tan\:x\;-\;\sec^{3}\:x\:dx\;+\;\int\:\sec\:x\:dx}}}$

On transposing the like terms,

$\\\;\displaystyle{\sf{\rightarrow\;\;\int\:\sec^{3}\:x\:dx\;+\;\sec^{3}\:x\:dx\;=\;\bf{\sec\:x\;\tan\:x\;+\;\int\:\sec\:x\:dx}}}$

$\\\;\displaystyle{\sf{\rightarrow\;\;2\bigg(\int\:\sec^{3}\:x\:dx\bigg)\;=\;\bf{\sec\:x\;\tan\:x\;+\;\int\:\sec\:x\:dx}}}$

On integrating,

$\\\;\displaystyle{\sf{\rightarrow\;\;2\bigg(\int\:\sec^{3}\:x\:dx\bigg)\;=\;\bf{\sec\:x\;\tan\:x\;+\;In\big|\sec\:x\;+\;\tan\:x\big|\;+\;c_{1}}}}$

$\\\;\displaystyle{\sf{\rightarrow\;\;\int\:\sec^{3}\:x\:dx\;=\;\bf{\bigg(\dfrac{\sec\:x\;\tan\:x\;+\;In\big|\sec\:x\;+\;\tan\:x\big|\;+\;c_{1}}{2}\bigg)}}}$

$\\\;\displaystyle{\sf{\rightarrow\;\;\int\:\sec^{3}\:x\:dx\;=\;\bf{\dfrac{\sec\:x\;\tan\:x}{2}\;+\dfrac{\;In\big|\sec\:x\;+\;\tan\:x\big|}{2}\;+\;\dfrac{c_{1}}{2}}}}$

$\\\;\displaystyle{\sf{\rightarrow\;\;\int\:\sec^{3}\:x\:dx\;=\;\bf{\red{\dfrac{1}{2}(\sec\:x\;\tan\:x)\;+\dfrac{1}{2}(In\big|\sec\:x\;+\;\tan\:x\big|)\;+\;C}}}}$

$\\\;\underline{\boxed{\tt{Required\:\;Integration\;=\;\bf{\purple{\dfrac{1}{2}(\sec\:x\;\tan\:x)\;+\dfrac{1}{2}(In\big|\sec\:x\;+\;\tan\:x\big|)\;+\;C}}}}}$

Answer 2

Step-by-step explanation:

Here the concept of Integration by parts has been used. We see we are given an expression to integrate. Firstly we split the expression into two parts. Then we will integrate both parts separately by rule of integration to find the answer.

Let's do it !!