Question

integral (5tanx-2cotx)^2dx​

Answer 1

I= integ.of (5tanx-2 cotx)^2.dx.

I=integ.of(25tan^2x-20.tanx.cotx+4.cot^2x).dx

I=integ.of [25(sec^2x-1)-20+4(cosec^2x-1)].dx

I=integ.of [25sec^2x-25–20+4cosec^2x-4].dx

I=integ.of[25sec^2x+4cosec^2x-49].dx

I=25tan x-4cot x -49x+C . ,Answer

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Answer 2

The value of the integral is $\int\limits{(5tanx-2cotx)^2} \, dx=25tan x-4cot x -49x+C$ where C is the integration constant.

Given,

An expression: $(5tanx-2cotx)^2$.

To Find,

The value of the integral: $\int\limits{(5tanx-2cotx)^2} \, dx$.

Solution,

The method of finding the value of the integral is as follows -

We will simplify the given expression.

$(5tanx-2cotx)^2$$=25tan^2x-20.tanx.cotx+4.cot^2x$

$=25(sec^2x-1)-20+4(cosec^2x-1)$ [Since $tanx.cotx=1$]

$=25sec^2x-25-20+4cosec^2x-4= 25sec^2x+4cosec^2x-49$

Now we will compute the value of the integral.

$\int\limits{(5tanx-2cotx)^2} \, dx= \int\limits {(25sec^2x+4cosec^2x-49)} \, dx$

$=25tan x-4cot x -49x+C$, where C is the integration constant.

Hence, the value of the integral is $\int\limits{(5tanx-2cotx)^2} \, dx=25tan x-4cot x -49x+C$ where C is the integration constant.

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