Evaluate the integral {eq}\displaystyle \int \left( y^{\frac{1}{2}} - y^2+\frac{1}{y^3} \right) \ dy {/eq}.

Question:

Evaluate the integral {eq}\displaystyle \int \left( y^{\frac{1}{2}} - y^2+\frac{1}{y^3} \right) \ dy {/eq}.

Power Rule

We know from the fundamental theorem of calculus that integration is the inverse of differentiation. And so we want to find the functions whose derivatives are the functions in our integrand. Note that in our at the grow all of our functions are power functions. Recall that the power rule says

{eq}\begin{align*} \frac{d}{dx} \ x^n &= nx^{n-1} \end{align*} {/eq}

And so in reverse, we can write this as

{eq}\begin{align*} \int x^n\ dx &= \frac1{n+1} x^{n+1} + C \end{align*} {/eq}

Answer and Explanation: 1

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We just want to apply the power rule in reverse here. Note that the last term in our integrand can be written as {eq}\frac1{y^3} = y^{-3} {/eq}. We...

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Basic Calculus: Rules & Formulas

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Chapter 3 / Lesson 6
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In this lesson, learn what basic calculus is. Moreover, discover the differential and integral calculus formulas and learn how to solve basic calculus problems with examples.


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