INTEGRATION OF SECX | INTEGRATION OF SEC^2X

INTEGRATION OF SECX | INTEGRATION OF SEC^2X

Integration of secx :

Integration of sec x is so important for beginner students, because Integration of secx does learn to you about the how to do separation and how to do operation on trigonometric function for integrate various type of function.

In this session we will try to evaluate the Integration of sec x ; ​\( \int secxdx \)

So, students lets starts the solving, this is the so simple method to solve it, just in there have a some separation on secx.

Integration of secx

We get,

Integration of secx

Now multiplying secx with the numerator ( secx + tanx ) and separate them

And we get,

Integration

let secx + tanx = z

now differentiating both side with the respect to x

we get,

Now putting the value and we get,

[where  c is Integral Constant]

Hence,

I = log|secx + tanx|+ c

Further you can separate that answer or solution you also get,

I = log|secx + tanx|+ c

Now replacing secx by 1/cosx and tanx by sinx/cosx

we get,

integration of secx and tanx

[where  c is Integral Constant]

Now dividing with the cos(x/2) on both of numerator and denominator

We get,

tanx seperation in integration

This is the final answer of the Integration of sec x. As your according you can use both of the answers.

And with the same way you can evaluate the integration of cosecx.

Integration of secx tanx

Now we will try to evaluate the integration of sec x tanx

\( \int \)​secx.tanx dx

Assume, secx = z

Now differentiating on both side with respect to x

We get,

derivatives of sex respect to x

 

 

 

Or, secx.tanx dx = dz

Now putting the value and we get,

\( \int(dz) \)

hence, I = z + c

Or, I = secx + c

Alternative method,

I =\[ \int secx.tanx dx \]

assume cosx = z

now differentiating on both side with respect to x

and we get,

-sinx dx = dz

Now putting the value on given integration

 

[ Where c is  integral constant ]

integration of sec x tan x

Hence,     I = secx + c

So students this is final answer of integration of sec^x tan^x and as you can see in both of the method answer will be same is secx + c

Integration of sec^2x [ integration of secant squared x ]

Let try to solving, in this case we solve the ​\( \int(sec^2x) dx \)
Once again, I = \( \int(sec^2x) dx \)

Now multiply tanx/tanx with the sec^2x

And we get,

Assume that tanx = z

Now differentiating on both side with respect to x

sec2x dx = dz

integration of sec^2x

Hence, I = tanx + c

So, tanx + c is the answer of integration of secant squared x or sec^2x.

 

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