How do you integrate $\int \sqrt{x^{3} + x^{2}} (3 x^{2} + 2 x)$ $int sqrt(x^3+x^2)(3x^2+2x)$ using substitution?

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How do you integrate $\int \sqrt{x^{3} + x^{2}} (3 x^{2} + 2 x)$ $int sqrt(x^3+x^2)(3x^2+2x)$ using substitution?

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Answer 1 · 2018-03-03T02:57:57

$\int \sqrt{x^{3} + x^{2}} (3 x^{2} + 2 x) d x = \frac{2}{3} {(x^{3} + x^{2})}^{\frac{3}{2}} + C$ $intsqrt(x^3+x^2)(3x^2+2x)dx=2/3(x^3+x^2)^(3/2)+C$

Explanation:

Let's set $u = x^{3} + x^{2} .$ $u=x^3+x^2.$

Differentiating, we get

$\frac{d u}{d x} = 3 x^{2} + 2 x$ $(du)/dx=3x^2+2x$

We can change this derivative to a differential by moving $d x$ $dx$ to the right side (multiplying both sides by it)

$d u = (3 x^{2} + 2 x) d x$ $du=(3x^2+2x)dx$

Look at our integral, $\int \sqrt{x^{3} + x^{2}} (3 x^{2} + 2 x) d x$ $intsqrt(x^3+x^2)(3x^2+2x)dx$

We see $(3 x^{2} + 2 x) d x$ $(3x^2+2x)dx$ appears in the integral; thus, we can replace it with $d u .$ $du.$ Furthermore, we already said $u = x^{3} + x^{2},$ $u=x^3+x^2,$ so we can replace everything under the root with $u .$ $u.$

$\int \sqrt{x^{3} + x^{2}} (3 x^{2} + 2 x) d x = \int \sqrt{u} d u$ $intsqrt(x^3+x^2)(3x^2+2x)dx=intsqrt(u)du$

Let's rewrite this with exponents:

$\int \sqrt{u} d u = \int u^{\frac{1}{2}} d u$ $intsqrt(u)du=intu^(1/2)du$

Recall that $\int x^{a} d x = \frac{x^{a + 1}}{a + 1} + C$ $intx^adx=x^(a+1)/(a+1)+C$ ; where $C$ $C$ is just the constant of integration; therefore,

$\int u^{\frac{1}{2}} d u = \frac{u^{\frac{1}{2} + 1}}{\frac{1}{2} + 1} + C = \frac{u^{\frac{3}{2}}}{\frac{3}{2}} + C = \frac{2}{3} u^{\frac{3}{2}} + C$ $intu^(1/2)du=u^(1/2+1)/(1/2+1)+C=u^(3/2)/(3/2)+C=2/3u^(3/2)+C$

Let's rewrite in terms of $x,$ $x,$ recalling that $u = x^{3} + x^{2} :$ $u=x^3+x^2:$

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

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How do you integrate $\int \sqrt{x^{3} + x^{2}} (3 x^{2} + 2 x)$ $int sqrt(x^3+x^2)(3x^2+2x)$ using substitution?

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How do you integrate $\int \sqrt{x^{3} + x^{2}} (3 x^{2} + 2 x)$ $int sqrt(x^3+x^2)(3x^2+2x)$ using substitution?