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U-Substitution, Integration by Parts and Differential Equations (8 Problems)

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Please show all of the steps needed to solve the 8 integrals and differential equations that are attached.

The integral of x(cos(x) dx
The integral of (x^3) sin(x) dx
The integral of t(csc(t))cot(t) dt
The integral of arctan x dx
The integral of e^2x sin(X) dx

Solve the differential equation.

y' = xe^x2

dy/dt = y^2/sqrt(2 + 3t)

cos(y) * y' = 2x.

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This solution shows how to calculate the integrals for the eight problems.

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