$\displaystyle\lim_{n\to\infty} \left(\frac{\left(n+1\right)^{\frac{1}{3}} }{n^{\frac{4}{3}}} + \frac{\left(n+2\right)^{\frac{1}{3}}}{n^{\frac{4}{3}}} + ..... + \frac{\left(2n\right)^{\frac{1}{3}}}{n^{\frac{4}{3}}}\right) $ equal to :
- $\frac{4}{3} \left(2\right)^{\frac{4}{3}} $
- $\frac{3}{4} \left(2\right)^{\frac{4}{3}} - \frac{4}{3} $
- $\frac{3}{4} \left(2\right)^{\frac{4}{3}} - \frac{3}{4} $
- $\frac{4}{3} \left(2\right)^{\frac{3}{4}} $
The Correct Option is C
Solution and Explanation
$ = \int^{1}_{0} \left(1+x\right)^{1/3} dx = \frac{3}{4} \left(2^{4/3} -1\right) $
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Concepts Used:
Integrals of Some Particular Functions
There are many important integration formulas which are applied to integrate many other standard integrals. In this article, we will take a look at the integrals of these particular functions and see how they are used in several other standard integrals.
Integrals of Some Particular Functions:
- ∫1/(x2 – a2) dx = (1/2a) log|(x – a)/(x + a)| + C
- ∫1/(a2 – x2) dx = (1/2a) log|(a + x)/(a – x)| + C
- ∫1/(x2 + a2) dx = (1/a) tan-1(x/a) + C
- ∫1/√(x2 – a2) dx = log|x + √(x2 – a2)| + C
- ∫1/√(a2 – x2) dx = sin-1(x/a) + C
- ∫1/√(x2 + a2) dx = log|x + √(x2 + a2)| + C
These are tabulated below along with the meaning of each part.
