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I have solved this integration question with two methods. I just want to know that which method is correct and why? As integration is reverse of derivation so when I try to differentiate both answers I get different functions. Itenter image description here must be same though.

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    Those functions look the same to me. What makes you think they are different? – 2017-01-04
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    @lulu +1 The only difference is that the constants of integration, which are unfortunately both called $C$, differ by $125/6$. – 2017-01-04
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    The only difference between two expressions is that the left one has one more summand $\;\frac{(-5)^3}6\;$ , but as indefinite integrals (or in its more accurate, and more and more widely used name: antiderivatives) is determined only up to a constant summand, both are correct. – 2017-01-04
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    Should get the same upon differentiation. Check your work. – 2017-01-04

3 Answers 3

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Expand the cube of the "left-side method" to see they are in fact the same; mind the constant of integration though.

Simpler example; on the one hand you have: $$\int x+1 \,\mbox{d}x = \frac{(x+1)^2}{2}+C=\color{green}{\frac{x^2}{2}+x}+\frac{1}{2}+\color{blue}{C}$$ But on the other hand: $$\int x+1 \,\mbox{d}x = \int x \,\mbox{d}x+\int 1 \,\mbox{d}x=\color{green}{\frac{x^2}{2}+x}+\color{red}{C}$$ Note that $\color{red}{C}=\frac{1}{2}+\color{blue}{C}$.

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    Now if you will differentiate answer then its different by 1/2. – 2017-01-04
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    The derivative of a constant (such as $\tfrac{1}{2}$) is $0$... – 2017-01-04
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    From 1st method after differentiating we get 1/2(2x-5)^2, while from 2nd method after differentiating we get exactly the same function that is given i.e (2x-5)^2 – 2017-01-04
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    But this 1/2 is not constant but coefficient here. – 2017-01-04
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    It _is_ a constant (term)... – 2017-01-04
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Note that $$\frac{1}{6}(2x-5)^3+c$$ $$=\frac{1}{6}(8x^3-60x^2+150x-125)+c$$ $$=\frac43x^3-10x^2+25x+(c-\frac{125}{6})$$ $$=\frac43x^3-10x^2+25x+C$$

Do you see the change in c and C?

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    Yes.. I understand this. What I really don't understand that from first or second method after differentiating I am getting different function. It suppose to be same – 2017-01-04
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    @Zonnie: The derivatives *are* the same: $$\left[\frac16(2x-5)^3\right]'=\frac16\cdot3(2x-5)^2\cdot2=(2x-5)^2=4x^2-20x+25\;,$$ and $$\left[\frac43x^3-10x^2+25x\right]'=\frac43\cdot3x^2-10\cdot2x+25=4x^2-20x+25\;.$$ – 2017-01-04
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Basically, you have the same answer, the only difference is the constant at the end of the solution.