Statement - 1 : The value of the integral `int(e^(3x)+e^(x))/(e^(4x)+1)dx " is "(1)/(sqrt(2))tan^(-1)((e^(x)-e^(-x))/(sqrt(2)))+C`
Statement -2: A primitive of the function f (x) `=(x^(2)+1)/(x^(4)+1)`is `(1)/(sqrt(2))tan^(-1)((x^(2)-1)/(sqrt(2)x))`.

Options

A. Statement - 1 True , Statement -2 is True , Statement -2 is a correct explanation for Statement -1.
B. Statement - 1 is True , Statement -2 is True , Statement -2 is a correct explanation for Statement -1.
C. Statement - 1 True ,Statement - 2 is False.
D. Statement - 1 is False , Statement - 2 is True.

Explanation

A primitive of the function f (x) `=(x^(2)+1)/(x^(4)+1)` is given by
`I=int(x^(2)+1)/(x^(4)+1)dx=int(1+(1)/(x^(2)))/(x^2+(1)/(x^(2)))dx`
`rArrI=int(1)/((x-(1)/(x))^(2)+(sqrt(2))^(2))d(x-(1)/(x))=(1)/(sqrt(2))tan^(-1)((x-(1)/(x))/(sqrt(2)))+C`
So , Statement - 2 is true.
Now , `I=int(e^(3x)+e^(x))/(e^(4x)+1)dx=int((e^(x))^(2)+1)/((e^(x))^(4)+1)d(e^(x))`
`rArrI=(1)/(sqrt(2))tan^(-1)((e^(2x)-1)/(sqrt(2)e^(x)))+C` " " [ Using statement -2]
`rArrI=(1)/(sqrt(2))tan^(-1)((e^(x)-e^(-x))/(sqrt(2)))+C`
So , statement - 1 is true .Also , statement -2 is a correct explanation for statement -1.