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Asked by prakashtarun313 | 13 May, 2020, 19:27: PM

Expert Answer

To find the integral of {1+(sinx)^2/3}/{1+(sinx)^4/3} w.r.t (sinx)^1/3

I equals integral fraction numerator 1 plus sin to the power of 2 divided by 3 end exponent x over denominator 1 plus sin to the power of 4 divided by 3 end exponent x end fraction d open parentheses cube root of sin x end root close parentheses equals integral fraction numerator 1 plus sin to the power of 2 divided by 3 end exponent x over denominator 1 plus sin to the power of 4 divided by 3 end exponent x end fraction open parentheses 1 third sin to the power of negative 2 divided by 3 end exponent x space cos x space d x close parentheses equals 1 third integral fraction numerator cos x open parentheses 1 plus sin to the power of 2 divided by 3 end exponent x close parentheses over denominator sin to the power of 2 divided by 3 end exponent x open parentheses 1 plus sin to the power of 4 divided by 3 end exponent x close parentheses end fraction d x T a k e space sin x equals t space rightwards double arrow space cos x space d x space equals space d t rightwards double arrow I equals 1 third integral fraction numerator open parentheses 1 plus t to the power of 2 divided by 3 end exponent close parentheses over denominator t to the power of 2 divided by 3 end exponent open parentheses 1 plus t to the power of 4 divided by 3 end exponent close parentheses end fraction d t N o w space t a k e space t to the power of 1 divided by 3 end exponent equals v space rightwards double arrow 1 third t to the power of negative 2 divided by 3 end exponent d t equals d v rightwards double arrow I equals integral fraction numerator 1 plus v squared over denominator 1 plus v to the power of 4 end fraction d v equals integral fraction numerator 1 over denominator open parentheses v minus begin display style fraction numerator 1 over denominator square root of 2 end fraction end style close parentheses squared plus begin display style 1 half end style end fraction d v T a k i n g space square root of 2 v minus 1 equals u space rightwards double arrow space square root of 2 d v equals d u rightwards double arrow I equals square root of 2 integral fraction numerator 1 over denominator 1 plus u squared end fraction d u equals square root of 2 space tan to the power of negative 1 end exponent u R e s u b s t i t u t i n g space t h e space v a l u e s comma space w e space g e t I equals fraction numerator 1 over denominator square root of 2 end fraction open square brackets tan to the power of negative 1 end exponent open parentheses fraction numerator 2 cube root of sin x end root plus square root of 2 over denominator square root of 2 end fraction close parentheses plus tan to the power of negative 1 end exponent open parentheses fraction numerator 2 cube root of sin x end root minus square root of 2 over denominator square root of 2 end fraction close parentheses close square brackets

Answered by Renu Varma | 17 May, 2020, 12:16: PM

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