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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.

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.

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.

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.

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 $x^3 dx$.

These techniques are combined when the integrand can be split into a product where one factor simplifies upon differentiation (for integration by parts) and the other factor is integrable only via substitution. This often occurs with polynomials multiplied by composite functions like $\sqrt{ax^n+b}$.

Conditions: The integrand is a product of two distinct types of functions.; One part reduces nicely when differentiated.; The other part requires a u-substitution to find its antiderivative.

These techniques are combined when the integrand can be split into a product where one factor simplifies upon differentiation (for integration by parts) and the other factor is integrable only via substitution. This often occurs with polynomials multiplied by composite functions like $\sqrt{ax^n+b}$.

Conditions: The integrand is a product of two distinct types of functions.; One part reduces nicely when differentiated.; The other part requires a u-substitution to find its antiderivative.

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.