ex and ln(x) are inverse functions.
With this you can get 5x = eln(5^x)
Therefore you can anti-differentiate this to get eln(5^x)/(ln(5x))
Which equals 5x/ln(5x)
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(that weird integral or antiderivative sign) x^(-6/5) dx =-5*x^(-1/5)
x(pi+1)/(pi+1)
By antiderivative do you mean integral? If yes, integral x^1 dx= (x^2)/2
Powers of e are simple to integrate. The derivative of eu equals u'eu; inversely, the antiderivative of eu equals eu/u'. Therefore, the antiderivative of e1/-x equals (e1/-x)/{d/dx[1/-x]}. The derivative of 1/-x, which can also be expressed as x-1, equals (-1)x(-1-1) = -x-2 = -1/x2.
I assume you mean -10x^4? In that case, antiderivative would be to add one to the exponent, then divide by the exponent. So -10x^5, then divide by 5. So the antiderivative is -2x^5.