Integration Questions: Check Answers and Simplify Solutions | Homework Help

[roadworx]

Homework Statement

I'm wondering if anyone can check these integrations for me, or suggest alternative answers if they're not quite right, or can be simplified?

1) $$\int x^{\sqrt{2}}$$

2) $$\int x . \sqrt{x}$$

3) $$\int \frac{1}{x^\pi}$$

Homework Equations

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The Attempt at a Solution

1) $$\int x^{1.4} = \frac{x^{2.4}}{2.4} + C$$

2) $$\int x . x^{\frac{1}{2}} = \frac{x^2}{2} . \frac{x^{\frac{3}{2}}}{\frac{3}{2}} = \frac{x^{\frac{3}{2}}}{3} . x^2 + C$$

3) $$\int \frac{1}{x^\pi} = \int x^{-\pi} = \frac{x^{-\pi+1}}{-\pi+1} + C$$

 

[Vuldoraq]

The first and the third look okay to me, although in the first one I would leave the $$\sqrt{2}$$ rather than putting 2.4.

In the second your almost there. When doing this sort of integral it is often easier if you simplify the integrand. By combining $$x*\sqrt{x}$$ into $$\sqrt{x*x^2}$$ to get $${x^{3/2}}$$ it should be easier to integrate.

 

[HallsofIvy]

Homework Statement

I'm wondering if anyone can check these integrations for me, or suggest alternative answers if they're not quite right, or can be simplified?

1) $$\int x^{\sqrt{2}}$$

2) $$\int x . \sqrt{x}$$

3) $$\int \frac{1}{x^\pi}$$

Homework Equations

--

The Attempt at a Solution

1) $$\int x^{1.4} = \frac{x^{2.4}}{2.4} + C$$

$\sqrt{2}$ is NOT equal to 1.4! There is no reason to change $x^{\sqrt{2}}$ to $x^{1.4}$.

2) $$\int x . x^{\frac{1}{2}} = \frac{x^2}{2} . \frac{x^{\frac{3}{2}}}{\frac{3}{2}} = \frac{x^{\frac{3}{2}}}{3} . x^2 + C$$

An unfortunately common mistake: just as you cannot differentiate a product by just differentiating each part, you cannot integrate a product that way either. As Vuldoraq said, $x\sqrt{x}= x(x^{1/2})= x^{3/2}$. Integrate that.

3) $$\int \frac{1}{x^\pi} = \int x^{-\pi} = \frac{x^{-\pi+1}}{-\pi+1} + C$$

Yes, this is correct.

 

[roadworx]

Thanks for the replies Vuldoraq, HallsofIvy.

So, my answer should be $$\int x^{\frac{3}{2}} = \frac{2x^{\frac{3}{2}}}{3}$$

 

[SunGod87]

You forgot to increase the exponent by one before dividing!

 

[dirk_mec1]

Are you aware of the fact that you're leaving out the dx in the integrals?

 

[roadworx]

alright..

Second attempt.

$$\int x^{\frac{3}{2}} = \frac{2x^{\frac{5}{2}}}{5}$$

 

[Vuldoraq]

Thats it, much better.

To repeat what HallsofIvy said: you can't integrate products of the function being integrated in the usual manner. So you have to simplify and when you can't simplify you have to use a different method (like integration by parts or substitution, you'll come across these later on).

May your Math prosper