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What are some good integrals??

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- Thread starter QuarkCharmer
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- #1

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What are some good integrals??

- #2

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What are some good integrals??

You may hit the SEARCH of this forum with 'integrals'.

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https://www.physicsforums.com/showpost.php?p=3433157&postcount=272

[tex] \int \sin(\ln x) + \cos(\ln x)dx[/tex]

[tex] \int \frac{x^2}{x^2 +4x + 8} dx[/tex]

[tex] \int \frac{1}{\sqrt{5x-3}+\sqrt{5x+2}} dx[/tex]

[tex] \int \left( x^2 + 1\right) e^{x^2}dx[/tex]

[tex] \int \frac{1}{\sqrt[3]{x} + x} dx[/tex]

The integral below is tricky, BUT it can be solved using only simple substitutions.

Show that

[tex] I_4 \, = \, \int_{0}^{\infty} \dfrac{x^{29}}{(5x^2+49)^{17}} \, dx \,=\, \dfrac{14!}{2\cdot 49^2 \cdot 5^{15 }\cdot 16!}[/tex]

What I like about these integrals, is that most of them have simple, clever solutions.

[tex] \int \sin(\ln x) + \cos(\ln x)dx[/tex]

[tex] \int \frac{x^2}{x^2 +4x + 8} dx[/tex]

[tex] \int \frac{1}{\sqrt{5x-3}+\sqrt{5x+2}} dx[/tex]

[tex] \int \left( x^2 + 1\right) e^{x^2}dx[/tex]

[tex] \int \frac{1}{\sqrt[3]{x} + x} dx[/tex]

The integral below is tricky, BUT it can be solved using only simple substitutions.

Show that

[tex] I_4 \, = \, \int_{0}^{\infty} \dfrac{x^{29}}{(5x^2+49)^{17}} \, dx \,=\, \dfrac{14!}{2\cdot 49^2 \cdot 5^{15 }\cdot 16!}[/tex]

What I like about these integrals, is that most of them have simple, clever solutions.

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- #4

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[tex]\int\frac{4x^5-1}{(x^5+x+1)^2}dx[/tex]

Once you see the solution of this one, you immediately get it. But without seeing the solution, it can be quite hard.

I'd suggest getting Apostol's calculus book. It is filled with hard integrals.

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