[itex]\int(x^{2}-5)^{2}x dx[/itex](adsbygoogle = window.adsbygoogle || []).push({});

By substiution:

1. [itex]u = x^{2}-5[/itex]

2. [itex]du = 2x dx[/itex]

3. [itex]\frac{du}{2x}= dx[/itex]

4. [itex]\int u^{2}x \frac{du}{2x}[/itex]

5. [itex]\int u^{2} \frac{1}{2} du[/itex]

6. [itex] \frac{1}{3} u^{3} \frac{1}{2}[/itex]

7. [itex] \frac{1}{6} u^{3}[/itex]

8. [itex] \frac{1}{6} (x^{2}-5)^{3}[/itex]

9. [itex] \frac{1}{6} [x^{6} - 15x^{4} + 75x^{2} +125][/itex]

By normal integration factorize the from beginning

from: [itex]\int(x^{2}-5)^{2}x dx[/itex]

to [itex]\int [x^{4} - 10x^{2} + 25 ] x dx[/itex]

then: [itex]\int x^{5} - 10x^{3} + 25x dx[/itex]

and finally : [itex] \frac{1}{6} [x^{6} - 15x^{4} + 75x^{2}] + C[/itex]

Probably this is a easy one, i been looking on internet, but had hard time to find the right keywords for an explanation...

Question is: the one give me some kind of constant and the other i just add one.. Which one is correct? I mean both gives same result except of one provide a "real" constant value.

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# Integration by subsitution (give constant value), why?

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