.....

Answer is 40

and also........... f(x)=e^x-e^-3x/2,the value of ₀∫^e2-e-3f^-1(x)dx is

The primes that divide 10! are 2,3,5 and 7 and we know how to compute the highest power of a prime that divides n!

Hence 10! = 2⁸3⁴5²7

Now you can check that
(3m+2)(3m+2) gives 3m+1
(3m+1)(3m+1) gives 3m+1
(3m+1)(3m+2) gives 3m+2

Now the problem requires us to ignore factors with powers of 2 and 3

5 is of the form 3m+2 while 7 is of the form 3m+1.

Hence 5 is one such factor, and the other is 35

5+35 = 40

for the integral put x = f(t). So the limits become 0 to 2 and the integrand becomes t(e^t + 3/2 e^-3t/2). This is easy to integrate by parts