Evaluate: $integral subscript 2 superscript 3 space 3 to the power of straight x space dx$ .

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Solution

straight I space equals space integral subscript 2 superscript 3 3 to the power of straight x space equals space fraction numerator 3 to the power of straight x over denominator log space 3 end fraction vertical line subscript 2 superscript 3 space plus space straight C space open parentheses Use colon space integral straight a to the power of straight x space equals space fraction numerator straight a to the power of straight x over denominator log space straight a end fraction space plus space straight C close parentheses space equals space 3 cubed over log minus fraction numerator 3 squared over denominator log space 3 end fraction space plus straight C equals space 1 over log cubed left parenthesis 3 cubed minus 3 squared right parenthesis space plus space straight C equals space 1 over log cubed left parenthesis 27 minus 9 right parenthesis space plus space straight C equals 1 over log cubed left parenthesis 18 right parenthesis space plus space straight C

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