How do you integrate $\int \frac{4 x}{\sqrt{3 x}}$ $int (4x)/sqrt(3x)$ using substitution?

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How do you integrate $\int \frac{4 x}{\sqrt{3 x}}$ $int (4x)/sqrt(3x)$ using substitution?

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Answer 1 · 2016-08-01T18:15:15

I found: $\frac{8}{27} {(3 x)}^{\frac{3}{2}} + c$ $8/27(3x)^(3/2)+c$

Explanation:

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Answer 2 · 2016-08-01T18:42:11

$\frac{8 x^{\frac{3}{2}}}{3^{\frac{3}{2}}} + C$ $(8x^(3/2))/3^(3/2)+C$

Explanation:

However, we should see that substitution is a waste of time:

$\int \frac{4 x}{\sqrt{3 x}} d x = \frac{4}{\sqrt{3}} \int \frac{x}{\sqrt{x}} d x = \frac{4}{\sqrt{3}} \int x^{\frac{1}{2}} d x$ $int(4x)/sqrt(3x)dx=4/sqrt3intx/sqrtxdx=4/sqrt3intx^(1/2)dx$

Using the rule $\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C$ $intx^ndx=x^(n+1)/(n+1)+C$ . where $n \neq - 1$ $n!=-1$ , this becomes:

$= \frac{4}{\sqrt{3}} (\frac{x^{\frac{3}{2}}}{\frac{3}{2}}) + C = \frac{8}{3 \sqrt{3}} x^{\frac{3}{2}} + C = \frac{8 x^{\frac{3}{2}}}{3^{\frac{3}{2}}} + C$ $=4/sqrt3(x^(3/2)/(3/2))+C=8/(3sqrt3)x^(3/2)+C=(8x^(3/2))/3^(3/2)+C$

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How do you integrate $\int \frac{4 x}{\sqrt{3 x}}$ $int (4x)/sqrt(3x)$ using substitution?

2 Answers

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How do you integrate $\int \frac{4 x}{\sqrt{3 x}}$ $int (4x)/sqrt(3x)$ using substitution?