How do you use substitution to integrate #4x( x^2 + 3)^4 dx#?

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To integrate (4x(x^2 + 3)^4), you can use the substitution method. Let (u = x^2 + 3). Then, (du/dx = 2x). Rearrange to solve for (x), obtaining (x = \sqrt{u - 3}). Now, replace (x) and (dx) in the integral with expressions in terms of (u). Substituting, (4x(x^2 + 3)^4 dx) becomes (4(\sqrt{u - 3})u^4 du). Now, integrate with respect to (u) and revert back to the original variable, (x).

To integrate ( 4x(x^2 + 3)^4 ) using substitution, let ( u = x^2 + 3 ). Then, ( du = 2x dx ). Rearrange ( du = 2x dx ) to solve for ( dx ), giving ( \frac{du}{2x} = dx ). Substitute ( u = x^2 + 3 ) and ( dx = \frac{du}{2x} ) into the integral. After substitution, the integral becomes ( \int 2u^4 du ). Integrate ( 2u^4 ) with respect to ( u ) to get ( \frac{2}{5}u^5 + C ), where ( C ) is the constant of integration. Finally, substitute back ( u = x^2 + 3 ) into the result to obtain the final integral expression.