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Using the quotient rule: Let u=x and v=2x+5 then du/dx= 1 and dv/dx=2 Hence dy/dx=[(2x+5)x1- (2x)]/(2x+5)²             =5/(2x+5)²

Integrate by parts: integral = [uv] - ∫u'v dx (u'= derivative of u, v'= derivative of v)

u= x     u'= 1

v' = sin2x        v= -0.5cos2x

= -0.5xcosx  -  ∫-0.5cos2x dx

= -0.5xcosx...

Integrate, y= 3x² -1/x

 1{³ 3x² - 1/x dx = [x-lnx]³₁= (3-ln3)-(1-ln1) = 3-ln3-1+0= 2-ln3

f(1)=0

f(x)=(x-1)(x+4)(x-2)

We are given: y = 4x^3  – 5/(x^2) To find the dy/dx we are going to use the power rule, from the power rule differentiating x^n gives n*x^n-1, so from our equation differetiating x^3 will give 3x^2, but w...