int(10x^(3)-5x)/(sqrt(x^(4)-x^(2)+6))dx

Given Integral: We are given the integral: int((10x^3 - 5x) / sqrt(x^4 - x^2 + 6))dx To solve this integral, we will first look for a substitution that simplifies the integrand. A natural choice is to let u be the expression under the square root in the denominator. Let u = x^4 - x^2 + 6.Substitution: Now, we differentiate u with respect to x to find du: du/dx = 4x^3 - 2x du = (4x^3 - 2x)dxDifferentiation: We notice that the numerator of the integrand, 10x^3 - 5x, is not a multiple of du. However, we can factor out a 5 from the numerator to see if it helps: 5(2x^3 - x)dx We can now see that if we multiply du by 5/2, we get the numerator of the integrand: (5/2)du = 5(2x^3 - x)dxFactor Out and Match: Now we can express the integral in terms of u: int((10x^3 - 5x) / sqrt(x^4 - x^2 + 6))dx = int((5/2)du / sqrt(u))Express in terms of u: Simplify the integral: int((5/2)du / sqrt(u)) = (5/2) * int(1 / sqrt(u))duSimplify the Integral: The integral of 1 / sqrt(u) with respect to u is 2 * sqrt(u), so we can integrate: (5/2) * int(1 / sqrt(u))du = (5/2) * 2 * sqrt(u) + CIntegrate 1/sqrt(u): Simplify the expression: (5/2) * 2 * sqrt(u) + C = 5 * sqrt(u) + CFinal Simplification: Now we substitute back the original expression for u: 5 * sqrt(u) + C = 5 * sqrt(x^4 - x^2 + 6) + CSubstitute Back: We have found the indefinite integral: int((10x^3 - 5x) / sqrt(x^4 - x^2 + 6))dx = 5 * sqrt(x^4 - x^2 + 6) + C

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\int\frac{10x^{3}-5x}{\sqrt{x^{4}-x^{2}+6}}\,dx

Given Integral: We are given the integral: $\newline$ $\int\frac{10x^3 - 5x}{\sqrt{x^4 - x^2 + 6}}dx$ $\newline$ To solve this integral, we will first look for a substitution that simplifies the integrand. A natural choice is to let $u$ be the expression under the square root in the denominator. $\newline$ Let $u = x^4 - x^2 + 6$ .
Substitution: Now, we differentiate $u$ with respect to $x$ to find $du$ : $\frac{du}{dx} = 4x^3 - 2x$ $du = (4x^3 - 2x)dx$
Differentiation: We notice that the numerator of the integrand, $10x^3 - 5x$ , is not a multiple of $du$ . However, we can factor out a $5$ from the numerator to see if it helps: $\newline$ $5(2x^3 - x)dx$ $\newline$ We can now see that if we multiply $du$ by $5/2$ , we get the numerator of the integrand: $\newline$ $(5/2)du = 5(2x^3 - x)dx$
Factor Out and Match: Now we can express the integral in terms of $u$ : $\int\left(\frac{10x^3 - 5x}{\sqrt{x^4 - x^2 + 6}}\right)dx = \int\left(\frac{5}{2}\frac{du}{\sqrt{u}}\right)$
Express in terms of $u$ : Simplify the integral: $\newline$ $\int\left(\frac{5}{2}\right)\frac{du}{\sqrt{u}} = \left(\frac{5}{2}\right) \cdot \int\frac{1}{\sqrt{u}}du$
Simplify the Integral: The integral of $\frac{1}{\sqrt{u}}$ with respect to $u$ is $2 \sqrt{u}$ , so we can integrate: $\newline$ $(\frac{5}{2}) \int \frac{1}{\sqrt{u}}du = (\frac{5}{2}) \cdot 2 \sqrt{u} + C$
Integrate $\frac{1}{\sqrt{u}}$ : Simplify the expression: $\newline$ $\frac{5}{2} \times 2 \times \sqrt{u} + C = 5 \times \sqrt{u} + C$
Final Simplification: Now we substitute back the original expression for $u$ : $5 \times \sqrt{u} + C = 5 \times \sqrt{x^4 - x^2 + 6} + C$
Substitute Back: We have found the indefinite integral: $\newline$ $\int\left(\frac{10x^3 - 5x}{\sqrt{x^4 - x^2 + 6}}\right)dx = 5 \cdot \sqrt{x^4 - x^2 + 6} + C$