Evaluate the following integral using substitution.
$integral subscript 0 superscript 2 straight x space square root of straight x plus 2 end root dx.$

Solution

Let I = $integral subscript 0 superscript 2 straight x square root of straight x plus 2 end root dx$
Put $square root of straight x plus 2 end root space equals space straight y space space or space space straight x plus 2 space equals space straight y squared space space space space space space rightwards double arrow space space space space straight x space equals space straight y squared minus 2$
$therefore space space space space dx space equals space 2 space straight y space dy$
When x = 0, $straight y equals square root of 0 plus 2 end root space equals space square root of 2$
When x = 2, $straight y space equals space square root of 2 plus 2 end root space equals space square root of 4 space equals space 2$
$therefore space space space space straight I space equals space integral subscript square root of 2 end subscript superscript 2 space left parenthesis straight y squared minus 2 right parenthesis. space straight y space. space 2 space straight y space dy space equals space 2 integral subscript square root of 2 end subscript superscript 2 space left parenthesis straight y squared minus 2 right parenthesis. space straight y squared space dy$
$equals space 2 integral subscript square root of 2 end subscript superscript 2 left parenthesis straight y to the power of 4 minus 2 straight y squared right parenthesis space dy space equals space 2 open square brackets straight y to the power of 5 over 5 minus 2 over 3 straight y cubed close square brackets subscript square root of 2 end subscript superscript 2$
$space equals space 2 open square brackets open parentheses 32 over 5 minus 16 over 3 close parentheses minus open parentheses fraction numerator 4 square root of 2 over denominator 5 end fraction minus fraction numerator 4 square root of 2 over denominator 3 end fraction close parentheses close square brackets space equals space 2 open square brackets fraction numerator 96 minus 80 over denominator 15 end fraction minus fraction numerator 12 square root of 2 minus 20 square root of 2 over denominator 15 end fraction close square brackets$
= $2 open square brackets 16 over 15 plus fraction numerator 8 square root of 2 over denominator 15 end fraction close square brackets space equals space 16 over 15 left parenthesis 2 plus square root of 2 right parenthesis$

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Integrals

Evaluate the following integral using substitution.
$integral subscript 0 superscript 2 straight x space square root of straight x plus 2 end root dx.$

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