What happens when integration by parts is applied twice to $e^x \cos x$?

LIATE suggests which factor to differentiate (assign to $u$) based on the order: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential. However, it is not an absolute rule.

Conditions: The integrand involves a product of functions from different LIATE categories.; The user is selecting $u$ and $dv$ for integration by parts.

The factor $x^3$ is grouped with $\sqrt{1+x^4}$ because the derivative of the inner function $1+x^4$ is $4x^3$. This presence of $x^3$ makes the expression $x^3\sqrt{1+x^4}$ directly integrable using substitution, which is a prerequisite for the integration by parts method.

Conditions: The integrand is $x^7\sqrt{1+x^4}$.; The method involves integration by parts.; The term $dv$ must be integrable.

The substitution provides the antiderivative $v$ needed for the integration by parts formula $\int u dv = uv - \int v du$. Without calculating $v$ via substitution, the integration by parts step cannot be completed.

Conditions: The integral requires integration by parts.; One part of the integrand ($dv$) requires substitution to integrate.; The other part ($u$) is easily differentiated.

Rewriting the integrand separates it into a factor suitable for differentiation and a factor suitable for integration. Choosing $x^4$ for differentiation reduces its power, while the remaining factor $x^3\sqrt{1+x^4}$ becomes integrable via substitution because its derivative contains the necessary $x^3$ term.

Conditions: The integrand is $x^7\sqrt{1+x^4}$.; Integration by parts is the chosen method.; A substitution is available for the second factor.

Integration by parts replaces an integral of $u \, dv$ with $uv$ minus an integral of $v \, du$. The goal is to choose factors that make the new integral easier to evaluate.

Conditions: The integrand can be expressed as a product of two functions.; The chosen factors $u$ and $dv$ must be differentiable and integrable respectively.