Evaluate the integral. int(x^(2)+1)/(sqrt(x^(3)+3x+20))dx.

Identify Integral: Identify the integral to be solved. We need to evaluate the integral of the function (x^2 + 1) / sqrt(x^3 + 3x + 20) with respect to x.Find Substitution: Look for a substitution that simplifies the integral. Let u = x^3 + 3x + 20, then du/dx = 3x^2 + 3. Notice that x^2 is part of the derivative of u, which suggests that this substitution might be useful.Express x^2: Express x^2 in terms of u and du. Since du/dx = 3x^2 + 3, we can solve for x^2: x^2 = (du/dx - 3)/3. We will use this expression to replace x^2 in the integral.Perform Substitution: Perform the substitution in the integral. Substitute u for x^3 + 3x + 20 and (du/dx - 3)/3 for x^2 in the integral. The integral becomes: ∫((du/dx - 3)/3 + 1) / sqrt(u) dxSimplify Integral: Simplify the integral. The integral simplifies to: ∫(1/3 du/dx - 1 + 1) / sqrt(u) dx = ∫(1/3 du/dx) / sqrt(u) dx = 1/3 ∫du / sqrt(u)Integrate with u: Integrate with respect to u. The integral of 1/sqrt(u) with respect to u is 2*sqrt(u), so we have: 1/3 * 2 * sqrt(u) + C = (2/3) * sqrt(u) + CSubstitute back: Substitute back for u. Replace u with x^3 + 3x + 20 to get the final answer: (2/3) * sqrt(x^3 + 3x + 20) + C

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Evaluate the integral. int(x^(2)+1)/(sqrt(x^(3)+3x+20))dx.

Evaluate the integral. $\int \frac{x^{2}+1}{\sqrt{x^{3}+3 x+20}} d x$ .

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Calculus

Evaluate definite integrals using the chain rule

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Q. Evaluate the integral.

\int \frac{x^{2}+1}{\sqrt{x^{3}+3 x+20}} d x

Identify Integral: Identify the integral to be solved. $\newline$ We need to evaluate the integral of the function $\frac{x^2 + 1}{\sqrt{x^3 + 3x + 20}}$ with respect to $x$ .
Find Substitution: Look for a substitution that simplifies the integral. $\newline$ Let $u = x^3 + 3x + 20$ , then $\frac{du}{dx} = 3x^2 + 3$ . Notice that $x^2$ is part of the derivative of $u$ , which suggests that this substitution might be useful.
Express $x^2$ : Express $x^2$ in terms of $u$ and $du$ . $\newline$ Since $\frac{du}{dx} = 3x^2 + 3$ , we can solve for $x^2$ : $x^2 = \frac{du/dx - 3}{3}$ . We will use this expression to replace $x^2$ in the integral.
Perform Substitution: Perform the substitution in the integral. $\newline$ Substitute $u$ for $x^3 + 3x + 20$ and $(\frac{du}{dx} - 3)/3$ for $x^2$ in the integral. The integral becomes: $\newline$ $\int\left(\frac{\frac{du}{dx} - 3}{3} + 1\right) / \sqrt{u} \, dx$
Simplify Integral: Simplify the integral. $\newline$ The integral simplifies to: $\newline$ $\int\left(\frac{1}{3} \frac{du}{dx} - 1 + 1\right) / \sqrt{u} \, dx$ $\newline$ = $\int\left(\frac{1}{3} \frac{du}{dx}\right) / \sqrt{u} \, dx$ $\newline$ = $\frac{1}{3} \int\frac{du}{\sqrt{u}}$
Integrate with u: Integrate with respect to $u$ . The integral of $\frac{1}{\sqrt{u}}$ with respect to $u$ is $2\sqrt{u}$ , so we have: $\frac{1}{3} \times 2 \times \sqrt{u} + C = \left(\frac{2}{3}\right) \times \sqrt{u} + C$
Substitute back: Substitute back for $u$ . Replace $u$ with $x^3 + 3x + 20$ to get the final answer: $\frac{2}{3} \cdot \sqrt{x^3 + 3x + 20} + C$