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Statement - 1 : The value of the integra...

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))`.

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.

Text Solution

AI Generated Solution

To solve the given integral \(\int \frac{e^{3x} + e^x}{e^{4x} + 1} \, dx\), we will follow these steps: ### Step 1: Simplify the Integral We start by factoring out \(e^x\) from the numerator and denominator: \[ \int \frac{e^{3x} + e^x}{e^{4x} + 1} \, dx = \int \frac{e^x(e^{2x} + 1)}{e^{4x} + 1} \, dx \] ...
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