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Since we have two functions of x being multipied togrther, we have to integrate this by parts. Therefore if we say, u=sinx and v'=e^x, then u'=cosx and v=e^x.Applying the integration b...

Integration by parts follows the general form: ∫u (dv/dx) = u.v - ∫v (du/dx)Let x = uLet sin 7x = (dv/dx)∫x.sin(7x) dx = x.(-1/7)cos(7x) - ∫(-1/7)cos(7x).1 dx = (-x/7)cos(7x) + (1/49)sin(7x) + c

3x^2 Integrate you add one to the power and divide by that number. In this case:3x^2 = (3x^3)/3 = x^3

First notice that f(x) = u/v. So f'(x) =[ v(u') - u(v')]/v² (the Quotient rule). After working it out, we find f'(x) = 6x/(x² + 5)² (the steps can be shown on the whiteboa...