SOLUTION: I have 2 indefinite integral problems I am struggling with, please help me if you can. #1. dx/3x^10=?+c #2. (3+4u)/u du =?

Question 534008: I have 2 indefinite integral problems I am struggling with, please help me if you can.
#1. dx/3x^10=?+c #2. (3+4u)/u du =?
Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
First problem:
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Bring the constant outside the integration sign.
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Recall that if a variable in the denominator has a positive exponent, it can be brought into the numerator with a negative exponent. So, the in the denominator can be moved into the numerator as changing the problem to:
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Now integrate using the rule
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This rule results in the integration becoming:
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Simplify the answer by adding the -10+1 in two places to get the answer:
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Further simplify the answer by factoring out the -9 from the denominator and multiplying it times the 3 in the denominator of the multiplier to make it:
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It would make things a little more conventional to move the variable x into the denominator by changing its exponent from -9 to +9 in doing so:
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Finally, since this was an indefinite integration, don't forget to include the constant c in the answer ... thereby making the final answer:
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You could also write this as:
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If you so choose to do so.
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Second problem:
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Begin splitting this into two separate problems by dividing the u into the denominator into both terms of the numerator as follows:
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Now multiply both of these two terms by du and you change the problem to:
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In the first integration move the constant 3 outside the integration sign and in the second integration move the constant 4 outside the integration sign. When you do those two moves you have:
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You should recognize the first integration is of the standard form
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Note that the x is an absolute value.
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Completing the first integration and multiplying it by the constant 3 results in the problem being reduced to:
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In the second integral notice that so you are left with the following:
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and the integration of du results in just u. Since this is problem involves just indefinite integrals the constant c must be added and the result becomes:
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That's it. Hope that these two problems help you to understand better the process of integration or anti-derivation, whatever you have been taught to call it.
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