Using u-substitution (where u = sinx), you'll find the antiderivative to be 0.5*sin2x + C.
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X(logX-1) + C
(2/3)*x^(3/2)
If in the real number universe, first k is to be >0, y=kx = exlog(k) the antiderivative of eax is eax/a so the antiderivative of Y is exlog(k) / log(k) = kx /log(k)
x/(x+1) = 1 - 1/(x + 1), so the antiderivative (or indefinite integral) is x + ln |x + 1| + C,
∫ 4/x dx= 4 ∫ 1/x dx= 4ln(x) + CThis is true for three reasons:the derivative of the term ln(x) is equal to 1/x4 is a constant factor of the term, and can be moved out of the integralC is an unknown constant, because we're looking at an indefinite integralYou can confirm this by taking the derivative of 4ln(x), which gives you 4/x, the original term.