What would you like to understand?

The substitution works because the derivative of $1+x^4$ is $4x^3$, which matches the $x^3$ term present in the integrand $x^3\sqrt{1+x^4}$. This allows the integral to be transformed into a simpler power function form, enabling the calculation of the antiderivative required for the integration by parts step.

Conditions: The term to be integrated is $x^3\sqrt{1+x^4}$.; The substitution variable is $w = 1+x^4$.; The differential $dw$ is proportional to $x^3 dx$.

The substitution works because the derivative of $1+x^4$ is $4x^3$, which matches the $x^3$ term present in the integrand $x^3\sqrt{1+x^4}$. This allows the integral to be transformed into a simpler power function form, enabling the calculation of the antiderivative required for the integration by parts step.

Conditions: The term to be integrated is $x^3\sqrt{1+x^4}$.; The substitution variable is $w = 1+x^4$.; The differential $dw$ is proportional to $x^3 dx$.

For $\int x e^x \, dx$, choose $u=x$ and $dv=e^x \, dx$. This gives $du=dx$ and $v=e^x$.

Conditions: The integrand is $x e^x$.; The method is integration by parts.

For $\int x e^x \, dx$, choose $u=x$ and $dv=e^x \, dx$. This gives $du=dx$ and $v=e^x$.

Conditions: The integrand is $x e^x$.; The method is integration by parts.