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We will solve this question with the knowledge that dy/dx = u.(dv/dx) + v.(du/dx), where y=u.v We have y=e^x(3x+1)^2. First, we want to find u & v. By splitting the function, we have that u=e^x and v=...

We have 2tan(th) / (1 + tan^2(th)) = sin(2th)

We know that tan(A) = sin(A) / cos(A), and 1 + tan^2(A) = sec^2(A)

Therefore => (2sin(th) / cos(th)) / sec^2(th)

=> 2sin(th)*cos^2(...

We have 3^(2x+1) = 4^100

=> log(3^(2x+1)) = log(4^100)

=> (2x+1)log(3) = 100log(4)