Seeking indefinite integral ∫ (xsinx) &# 178; DX

Seeking indefinite integral ∫ (xsinx) &# 178; DX

∫(xsinx)/(cosx)^3 dx

∫ (xsinx) / (cosx) ^ 3 DX = ∫ xtanx (secx) ^ 2 DX = ∫ xtanxdtanx = 1 / 2 ∫ XD (TaNx) ^ 2 = 1 / 2 [x (TaNx) ^ 2 - ∫ (TaNx) ^ 2DX]: ∫ (TaNx) ^ 2DX = ∫ [(secx) ^ 2-1] DX = tanx-x + C, so the original formula = (x (TaNx) ^ 2-tanx + x) / 2 + C

Finding definite integral ∫ 2x ^ 2E ^ (- 2x) DX (0 to positive infinity)
The exponent of X is 2, the exponent of E is - 2x, why is the result 1 / 2

Partial integral is ∫ f (x) DG (x) = f (x) g (x) - ∫ g (x) DF (x), so the original formula = - ∫ X & sup2; e ^ (- 2x) d (- 2x) = - ∫ X & sup2; de ^ (- 2x) = - [x & sup2; * e ^ (- 2x) - ∫ e ^ (- 2x) DX & sup2;] = - X & sup2; * e ^ (- 2x) + ∫ 2xe ^ (- 2x) DX = - X & sup2; * e ^ (- 2x) - ∫ Xe ^ (- 2x) d (- 2x) = - X