antidifferentiation

[Becky4paws]

I am having so much trouble with these problems...any explanation would be appreciated.

integral (square root x^3) - (1/2*square root x) + (square root 2)

= (x^3)^1/2 - 1/2*x^-1/2 + 2^1/2 dx
= (x^3)^3/2 - 1/2*x^1/2 + 2^3/2

Oh, please advise

 

[morson]

\(\displaystyle \ \int x^n\,dx\ = \frac{x^{n+1}}{n+1}\ + C\), \(\displaystyle n \not=\ -1\)

 

[soroban]

Hello, Becky4paws!

You're working with fractional exponents now,
. . yet you act like you've never seen the formula . . .


\(\displaystyle \L\int\left[\left(\sqrt{x}\right)^3 \,-\,\frac{1}{2}\sqrt{x} \;+\, \sqrt{2}\right]\,dx \;=\;\L\int\left(x^{\frac{3}{2}}\,-\,\frac{1}{2}x^{\frac{1}{2}}\,+\,\sqrt{2}\right)\,dx\)


Integrate: \(\displaystyle \L\frac{x^{\frac{5}{2}}}{\frac{5}{2}}\,-\,\frac{1}{2}\frac{x^{\frac{3}{2}}}{\frac{3}{2}} \,+\,\sqrt{2}\cdot x\,+\,C \;=\;\frac{2}{5}x^{\frac{5}{2}} \,-\,\frac{1}{3}x^{\frac{3}{2}} \,+\,\sqrt{2}\cdot x\,+\,C\)