∫ ln(2−3x) dx = x ln(2−3x) − ∫ −3x/(2−3x) dx
.................. = x ln(2−3x) − ∫ ((2−3x)/(2−3x) − 2/(2−3x)) dx
.................. = x ln(2−3x) − ∫ (1 − 2/(2−3x)) dx
.................. = x ln(2−3x) − (x + 2/3 ln(2−3x)) + C
.................. = x ln(2−3x) − x − 2/3 ln(2−3x) + C
.................. = ln(2−3x) (x − 2/3) − x + C
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6b)
Integrate by parts:
u = x ............ dv = secx tanx dx
du = dx ........... v = secx
∫ x secx tanx dx = x secx − ∫ secx dx
∫ x secx tanx dx = x secx − ln |secx + tanx| + C
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6d)
Integrate by parts:
u = sinx ................ dv = e^(2x) dx
du = cosx dx .......... v = 1/2 e^(2x)
∫ e^(2x) sinx dx = 1/2 e^(2x) sinx − ∫ 1/2 e^(2x) cosx dx
Integrate by parts again:
u = cosx ................ dv = 1/2 e^(2x) dx
du = −sinx dx .......... v = 1/4 e^(2x)
∫ e^(2x) sinx dx = 1/2 e^(2x) sinx − (1/4 e^(2x) cosx + ∫ 1/4 e^(2x) sinx dx)
∫ e^(2x) sinx dx = 1/2 e^(2x) sinx − 1/4 e^(2x) cosx − 1/4 ∫ e^(2x) sinx dx
∫ e^(2x) sinx dx + 1/4 ∫ e^(2x) sinx dx = 1/2 e^(2x) sinx − 1/4 e^(2x) cosx
5/4 ∫ e^(2x) sinx dx = 1/2 e^(2x) sinx − 1/4 e^(2x) cosx
∫ e^(2x) sinx dx = 4/5 (1/2 e^(2x) sinx − 1/4 e^(2x) cosx) + C
∫ e^(2x) sinx dx = 4/5 * 1/4 e^(2x) (2 sinx − cosx) + C
∫ e^(2x) sinx dx = 1/5 e^(2x) (2 sinx − cosx) + C