Integration and differentiation: int(x^3-6x^2+2x+5)dx, int(4sin(2*theta)d*theta

[jamescopeman]

Hey guys, first post. These two topics are extremely similar, was i right to include both in a single post?
Thanks in advance for any help you can provide.

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P10: Integration

. . .4. Integrate this function with respect to x.

. . . . .\(\displaystyle \displaystyle \int\, \left(x^3\, -\,6x^2\, +\, 2x\, +\, 5\right)\, dx\)

. . .5. Integrate this function with respect to \(\displaystyle \theta .\)

. . . . .\(\displaystyle \displaystyle \int\, 4\sin\left(2\theta\right)\, d\theta\)

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This integration is something i found hard when we were studying it, i'm revising and i just cant wrap my head around these two, could someone help me out with it? Only problem is its been so long i'm really struggling. I know it's cheeky but answers would be really helpful for me to figure out whats going on, and be something to aim for if that makes sense.

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. . . . .\(\displaystyle i \, =\, \dfrac{V}{R}\,\left(1\, -\, e^{-t/RC}\right)\)

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Don't have any real trouble with differentiation, have found every question but this one easy. any ideas why i cant do this correctly?

 

[HallsofIvy]

For problems 4 and 5, you need to use the basic integration fact that \(\displaystyle \int x^n dx= \frac{1}{n+1}x^{n+1}+ C\). Further to integrate a sum, do each term separately. For example, \(\displaystyle \int 5x^3- 3x+ 1 dx= \frac{5}{4}x^4- \frac{3}{2}x^2+ x+ C\).

For the last problem, you give an equation, \(\displaystyle i= \frac{V}{R}\left(1- e^{-t/RC}\right)\), but no question or problem! You mention "dfferentition" so I can guess that you want to differentiate i but with respect to which of the four variables in that function? I could again guess that the derivative is to be with respect to t. If that is correct then
1)\(\displaystyle \frac{V}{R}\) is constant with respect to t so we just multiply by the derivative of the rest.

2) The derivative of \(\displaystyle e^x\) is just \(\displaystyle e^x\) and, by the chain rule, the derivative of \(\displaystyle 1- e^{-t/RC}\) is \(\displaystyle 0- \left(\frac{-1}{RC}\right)e^{-t/RC}\).

So the derivative of \(\displaystyle i= \frac{V}{R}\left(1- e^{-t/RC}\right)\), with respect to t, is \(\displaystyle \frac{di}{dt}= \frac{V}{R}\left(\frac{1}{RC}\right)e^{-t/RC}= \frac{V}{R^2C}e^{-t/RC}\).

 

[mmm4444bot]

[Integration and differentiation] are extremely similar, was i right to include both in a single post?

Both operations are calculus topics, but I wouldn't label their similarity as "extreme".

We like to see separate threads for separate exercises, even when the exercises are from the same topic. Thanks. :cool:

 

[mmm4444bot]

...any ideas why i cant do this correctly?

No. But, if you were to show your work, we would have an idea. :cool: