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  • Evaluate the integrals in Exercises 1 to 8 using substitution. Integral 0 to 2 x root pf x + 2 ( Put x + 2 = t^2)

Evaluate the integrals in Exercises 1 to 8 using substitution.

    Q4.    \int_0^2x\sqrt{x+2}. (Put {x+2} = t^2)

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Let x+2 = t^2\Rightarrow dx =2tdt
when x = 0 then t = \sqrt{2} and when x=2 then t = 2

I=\int_{0}^{2}x\sqrt{x+2}dx

     \\=2\int_{\sqrt{2}}^{2}(t^2-2)t^2dt\\ =2\int_{\sqrt{2}}^{2}(t^4-2t^2)dt\\ =2[t^5/5-\frac{2}{3}t^3]^2_{\sqrt{2}}\\ =2[\frac{32}{5}-\frac{16}{3}-\frac{4\sqrt{2}}{5}+\frac{4\sqrt{2}}{3}]\\ =\frac{16\sqrt{2}(\sqrt{2}+1)}{15}

 

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manish

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