How do you integrate by substitution $\int \frac{x^{2} + 3 x + 7}{\sqrt{x}} d x$ $int (x^2+3x+7)/sqrtxdx$ ?

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Answer 1 · 2017-03-19T03:53:46

$(\frac{2}{5}) x^{\frac{5}{2}} + 2 x^{\frac{3}{2}} + 14 \sqrt{x} + C$ $(2/5)x^(5/2) +2x^(3/2) +14sqrtx +C$

Let u = x

$\int \frac{u^{2} + 3 u + 7}{\sqrt{u}}$ $int (u^2+3u +7)/sqrt(u)$

Make it easier by simplifying integrals

$\int \frac{u^{2}}{\sqrt{u}}$ $int u^2/sqrt(u)$ + $\int \frac{3 u}{\sqrt{u}}$ $int (3u)/sqrtu$ + $\int \frac{7}{\sqrt{u}}$ $int 7/sqrt(u)$

Simplify

$\int u^{\frac{3}{2}}$ $int u^(3/2)$ + $\int 3 \sqrt{u}$ $int 3sqrtu$ + $\int 7 u^{- \frac{1}{2}}$ $int 7u^(-1/2)$

Integrate

$\frac{2}{5} \cdot u^{\frac{5}{2}} + 3 \cdot (\frac{2}{3}) u^{\frac{3}{2}} + 7 \cdot 2 \sqrt{u}$ $2/5*u^(5/2) + 3*(2/3)u^(3/2) + 7*2sqrtu$

Sub back in x

$(\frac{2}{5}) x^{\frac{5}{2}} + 2 x^{\frac{3}{2}} + 14 \sqrt{x} + C$ $(2/5)x^(5/2) +2x^(3/2) +14sqrtx +C$

How do you integrate by substitution ∫x2+3x+7√xdxint (x^2+3x+7)/sqrtxdx?