the value of ∫\(\frac{dx}{x(1+log\,x)^3}\) is _______
- \(-\frac{x}{(1+log\,x)^2}\)\(+c\)
- \(-\frac{x}{(1+log\,x)^2}+c\)
- \(-\frac{1}{2(1+log\,x)^2}+c\)
- \(\frac{2}{(1+log\,x)^2}+c\)
The Correct Option is C
Solution and Explanation
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Concepts Used:
Integration by Parts
Integration by Parts is a mode of integrating 2 functions, when they multiplied with each other. For two functions ‘u’ and ‘v’, the formula is as follows:
∫u v dx = u∫v dx −∫u' (∫v dx) dx
- u is the first function u(x)
- v is the second function v(x)
- u' is the derivative of the function u(x)
The first function ‘u’ is used in the following order (ILATE):
- 'I' : Inverse Trigonometric Functions
- ‘L’ : Logarithmic Functions
- ‘A’ : Algebraic Functions
- ‘T’ : Trigonometric Functions
- ‘E’ : Exponential Functions
The rule as a diagram:
