 Hello Owen, how are you all today? My name is Priyanka and the question says integrate the following. Now here the function which is given to us is tan raised to the power 4 x. So we need to integrate this function with respect to x. Now here we can write this function as tan square x the whole rest of the power x into dx right. Also we know that tan square x is equal to secant square x minus 1. So it will be secant square x minus 1 whole rest of the power 2 into dx. Now on applying the formula that is a minus b the whole square we can write it as secant 4 x minus 2 secant square x plus 1 into dx. Now separating the integration sign we have integral secant 4 x dx minus taking out the constants to integral secant square x dx plus integral 1 dx. So we have now here if we put so now here let us simplify it we can write secant 4 x as secant square x whole rest of the power 2 dx minus 2 integral of secant square x dx is equal to tan x plus integral of 1 dx is x plus c 1. Let this be the first equation. Now let us simplify the integral of secant square x raised to the power 2 separately. Now we can write integral of secant square x raised to the power 2 dx equal to integral 1 plus tan square x whole rest of the power 2 dx right. Now 1 plus tan square x if not written as raised to the power 2 and multiplied by secant square x itself. So we can write the function as this also now if we put tan x equal to t we know that dt will be equal to secant square x dx right. So we have integral of 1 plus t square dt that is integral of 1 dt plus integral of t square dt which can be simplified as t plus tq by 3 plus c and on substituting the value of tan x back in t we have it equal to tan x plus tan q divided by 3 plus let it be c 2. So we can write in place of integral secant square x whole rest of the power 2 dx as equal to tan x plus tan q x upon 3 and writing down the further terms from here minus 2 tan x plus x plus c 1 plus c 2. So we have further simplified and written as 1 by 3 tan q x minus tan x plus x and if we combine c 1 and c 2 we can write it equal to c. So the required answer to the session is 1 upon 3 tan q x minus tan x plus c sorry plus x plus c. So this ends the session hope you understood the whole concept well have a nice day ahead.