Question

Integrate 3x^3 -2x+5/x√x with respect to x​

Answer 1

$\underline{\textsf{To find:}}$

$\mathsf{\displaystyle\int\;\dfrac{3x^3-2x+5}{x\sqrt{x}}\;dx}$

$\underline{\textsf{Solution:}}$

$\mathsf{Consider,}$

$\mathsf{\displaystyle\int\;\dfrac{3x^3-2x+5}{x\sqrt{x}}\;dx}$

$\mathsf{=\displaystyle\int\;\left[\dfrac{3x^3}{x\sqrt{x}}-\dfrac{2x}{x\sqrt{x}}+\dfrac{5}{x\sqrt{x}}\right]\;dx}$

$\mathsf{=\displaystyle\int\;\left[\dfrac{3x^2}{\sqrt{x}}-\dfrac{2}{\sqrt{x}}+\dfrac{5}{x\sqrt{x}}\right]\;dx}$

$\mathsf{=\displaystyle\int\;\left[3x^\frac{3}{2}-2x^\frac{-1}{2}+5x^\frac{-3}{2}\right]\;dx}$

$\mathsf{Using\;the\;formula}$

$\boxed{\mathsf{\int\,x^n\,dx=\dfrac{x^{n+1}}{n+1}+C}}$

$\mathsf{=\dfrac{3x^\frac{5}{2}}{\frac{5}{2}}-\dfrac{2x^\frac{1}{2}}{\frac{1}{2}}+\dfrac{5x^\frac{-1}{2}}{\frac{-1}{2}}+C}$

$\mathsf{=\dfrac{6x^\frac{5}{2}}{5}-4x^\frac{1}{2}-10x^\frac{-1}{2}+C}$

$\mathsf{=\dfrac{6x^2\sqrt{x}}{5}-4\sqrt{x}-\dfrac{10}{\sqrt{x}}+C}$

$\therefore\boxed{\mathsf{\displaystyle\int\;\dfrac{3x^3-2x+5}{x\sqrt{x}}\;dx=\dfrac{6x^2\sqrt{x}}{5}-4\sqrt{x}-\dfrac{10}{\sqrt{x}}+C}}$

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