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Maths-

General

Easy

Question

If f(x) is the primitive of $fraction numerator sin invisible function application root index 3 of x end root log invisible function application left parenthesis 1 plus 3 x right parenthesis over denominator left parenthesis tan to the power of negative 1 end exponent invisible function application square root of x right parenthesis to the power of 2 end exponent left parenthesis e to the power of root index 3 of x end root end exponent minus 1 right parenthesis end fraction$ (x $not equal to$ 0), then $stack l i m with x rightwards arrow 0 below f ´ left parenthesis x right parenthesis$ is-

0
3/5
5/3
None

The correct answer is: 3/5

f(x) = $not stretchy integral fraction numerator sin invisible function application x to the power of 1 divided by 3 end exponent log invisible function application left parenthesis 1 plus 3 x right parenthesis d x over denominator left parenthesis tan to the power of negative 1 end exponent invisible function application square root of x right parenthesis to the power of 2 end exponent left parenthesis e to the power of 5 x to the power of 1 divided by 3 end exponent end exponent minus 1 right parenthesis end fraction$
$stack l i m with x rightwards arrow 0 below f ´ left parenthesis x right parenthesis$ = $fraction numerator sin invisible function application x to the power of 1 divided by 3 end exponent log invisible function application left parenthesis 1 plus 3 x right parenthesis over denominator left parenthesis tan to the power of negative 1 end exponent invisible function application square root of x right parenthesis to the power of 2 end exponent left parenthesis e to the power of 5 x to the power of 1 divided by 3 end exponent end exponent minus 1 right parenthesis end fraction$
= $fraction numerator x to the power of 1 divided by 3 end exponent fraction numerator sin invisible function application x to the power of 1 divided by 3 end exponent over denominator x to the power of 1 divided by 3 end exponent end fraction. fraction numerator log invisible function application left parenthesis 1 plus 3 x right parenthesis over denominator 3 x end fraction.3 x over denominator x. open parentheses fraction numerator tan to the power of negative 1 end exponent invisible function application square root of x over denominator square root of x end fraction close parentheses to the power of 2 end exponent open parentheses fraction numerator e to the power of 5 x to the power of 1 divided by 3 end exponent end exponent minus 1 over denominator 5 x to the power of 1 divided by 3 end exponent end fraction close parentheses.5 x to the power of 1 divided by 3 end exponent end fraction$
= $fraction numerator 1.1.3 over denominator 1.1.5 end fraction equals$ 3/5

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