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sin 2x- 4e^(3x)...

`sin 2x- 4e^(3x)`

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Find an anti derivative (or integral) of the following functions by the method of inspection.sin2x-4e^(3x)

int e^(3x)(cos2x+sin2x)dx=e^(3x)*g(x)+c" Then "13[g(0)+g(pi/4)] =

int e^(3x)(cos2x+sin2x)dx=e^(3x)*g(x)+c Then 13[g(0)+g((pi)/(4))]=

Delta (x) = det [[sin2x, e ^ (x) sin x + x cos x, sin x + x ^ (2) cos xcos x + sin x, e ^ (x) + x, 1 + x ^ ( 2) e ^ (x) cos x, e ^ (2) x, e ^ (x)]]

Delta (x) = det [[sin2x, e ^ (x) sin x + x cos x, sin x + x ^ (2) cos xcos x + sin x, e ^ (x) + x, 1 + x ^ ( 2) e ^ (x) cos x, e ^ (2) x, e ^ (x)]]

e ^ (x) sin x tan x + e ^ (x) sin x + e ^ (x) sin x sec ^ (2) x

(sin x) ^ (cos x) + e ^ (3x)

Find number of real roots of equation e^(4x) + e^(3x) - 4e^(2x) + e^(x) + 1 = 0 is

int (2e ^ (5x) + e ^ (4x) -4e ^ (3x) + 4e ^ (2x) + 2e ^ (x)) / (e ((2x) +4) (e ^ (2x) - 1) ^ (2)) dx

int_(-pi//4)^(pi//4)(e^(x)x sin x)/(e^(2x)-1)dx=