Finding antiderivative problems

Question

I'm having trouble understanding how to find these antiderivatives:

1. $$e^2$$

Is the antiderivative just $e^2 \cdot x + c$

$$f(x) = \sqrt[3]{x^2} + x \sqrt{x}$$

$$= x^{2^{\frac{1}{3}}} + x \cdot x^{\frac{1}{2}} = x^{\frac{2}{3}} + x^{\frac{3}{2}}$$

So the antiderivative according to antiderivative formulas:

$$Anti = \frac{x^{\frac{5}{2}}}{\frac{5}{2}} + \frac{x^{\frac{5}{2}}}{\frac{5}{2}}$$

$$= 2 \cdot \frac{x^{\frac{5}{2}}}{\frac{5}{2}}$$

$$= 4 \cdot \frac{x^{\frac{5}{2}}}{5} + C = F$$

Is this right?

When I do a check, I don't get the original function:

$$F' = \frac{4}{5} \cdot \frac{5}{2} \cdot x ^{\frac{3}{2}} = 2 \cdot x^{\frac{3}{2}}$$

But that doesn't equal the original function. What did I do wrong?

Answer 1

1. Correct, because $e^2$ is just a constant, and for any constant function $f(x)=a$, the antiderivative is $ax+C$.

2. Note that $\frac{3}{2}\neq \frac{2}{3}$, and one becomes $\frac{5}{2}$, the other one becomes $\frac{5}{3}$, so the correct antiderivative is $\frac{2}{5}x^{\frac{5}{2}}+\frac{3}{5}x^{\frac{5}{3}}+C$.

Answer 2

1. It is correct
2. You are making a calculation mistake (considering $\frac 23$ as $ \frac32$) Otherwise all other things are correct

Answer 3

You have $$ x^{2^{\frac{1}{3}}} + x \cdot x^{\frac{1}{2}} = x^{\frac{2}{3}} + x^{\frac{3}{2}}$$ Now watch it!

$$Anti = \frac{x^{\frac{5}{3}}}{\frac{5}{3}} + \frac{x^{\frac{5}{2}}}{\frac{5}{2}}$$

That is why you do not get the right answer.