How do you integrate $\int x \sqrt{3 x + 1} d x$ $int xsqrt(3x+ 1) dx$ ?

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How do you integrate $\int x \sqrt{3 x + 1} d x$ $int xsqrt(3x+ 1) dx$ ?

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Answer 1 · 2017-04-15T16:05:01

$\int x \sqrt{3 x + 1} = \frac{2}{9} x {(3 x + 1)}^{\frac{3}{2}} - \frac{4}{135} {(3 x + 1)}^{\frac{5}{2}} + C$ $int xsqrt(3x + 1) = 2/9x(3x + 1)^(3/2) - 4/135(3x + 1)^(5/2) + C$

Explanation:

You will need substitution in addition to integration by parts. Let $u = x$ $u = x$ and $d v = \sqrt{3 x + 1}$ $dv = sqrt(3x +1)$ . By the power rule, $d u = d x$ $du = dx$ . But we will need to make a substitution to find $v$ $v$ . Let $t = 3 x + 1$ $t= 3x+ 1$ . Then $d t = 3 d x$ $dt = 3dx$ and $d x = \frac{d t}{3}$ $dx = (dt)/3$ .

$\int \sqrt{3 x + 1} = \sqrt{t} \cdot \frac{d t}{3} = \frac{1}{3} \sqrt{t} d t = \frac{2}{9} t^{\frac{3}{2}} = \frac{2}{9} {(3 x + 1)}^{\frac{3}{2}}$ $intsqrt(3x + 1) = sqrt(t) * (dt)/3 = 1/3sqrt(t)dt = 2/9t^(3/2) = 2/9(3x + 1)^(3/2)$

Apply integration by parts now.

$\int u d v = u v - \int v d u$ $int udv = uv - int vdu$

$\int x \sqrt{3 x + 1} = \frac{2}{9} x {(3 x + 1)}^{\frac{3}{2}} - \int \frac{2}{9} {(3 x + 1)}^{\frac{3}{2}} d x$ $intxsqrt(3x+ 1) = 2/9x(3x + 1)^(3/2) - int 2/9(3x+ 1)^(3/2)dx$

$\int x \sqrt{3 x + 1} = \frac{2}{9} x {(3 x + 1)}^{\frac{3}{2}} - \frac{2}{9} \int {(3 x + 1)}^{\frac{3}{2}} d x$ $int xsqrt(3x + 1) = 2/9x(3x + 1)^(3/2) - 2/9 int (3x + 1)^(3/2)dx$

Now let $n = 3 x + 1$ $n = 3x + 1$ . Then $d n = 3 d x$ $dn =3dx$ and $d x = \frac{d n}{3}$ $dx = (dn)/3$ .

$\int {(3 x + 1)}^{\frac{3}{2}} = \frac{1}{3} \int n^{\frac{3}{2}} d n = \frac{2}{15} n^{\frac{5}{2}} = \frac{2}{15} {(3 x + 1)}^{\frac{5}{2}}$ $int (3x + 1)^(3/2) = 1/3int n^(3/2) dn = 2/15n^(5/2) = 2/15(3x + 1)^(5/2)$

$\int x \sqrt{3 x + 1} = \frac{2}{9} x {(3 x + 1)}^{\frac{3}{2}} - \frac{2}{9} (\frac{2}{15}) {(3 x + 1)}^{\frac{5}{2}} + C$ $int xsqrt(3x + 1) = 2/9x(3x + 1)^(3/2) - 2/9(2/15)(3x +1)^(5/2) + C$

$\int x \sqrt{3 x + 1} = \frac{2}{9} x {(3 x + 1)}^{\frac{3}{2}} - \frac{4}{135} {(3 x + 1)}^{\frac{5}{2}} + C$ $int xsqrt(3x + 1) = 2/9x(3x + 1)^(3/2) - 4/135(3x + 1)^(5/2) + C$

Hopefully this helps!

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How do you integrate $\int x \sqrt{3 x + 1} d x$ $int xsqrt(3x+ 1) dx$ ?

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