Calculus - Techniques of Integration (Substitution, By Parts)

Integration by Substitution ($u$-substitution) is the reverse process of the Chain Rule. It is used when an integrand contains a function $g(x)$ and its derivative $g'(x)$. Visually, this technique can be viewed as a coordinate transformation that simplifies a complex area calculation on the $x$-axis into a simpler one on a $u$-axis, effectively 'straightening' the function's internal composition.

When applying substitution to definite integrals, it is essential to transform the limits of integration. If the original limits are $x = a$ and $x = b$, the new limits become $u = g(a)$ and $u = g(b)$. This ensures that the area under the curve is calculated over the correct interval in the transformed coordinate system without needing to substitute back to $x$.

Integration by Parts (IBP) is the integration equivalent of the Product Rule for differentiation. Geometrically, if you consider a curve in a $u$-$v$ plane, the formula relates the area under the curve $u(v)$ to the area of a rectangle $uv$ minus the area relative to the other axis. It is most effective when the integrand is a product of two dissimilar functions, such as an algebraic and a trigonometric function.

The LIATE rule is a visual mnemonic used to choose which part of the integrand should be assigned to $u$ in Integration by Parts. The priority order is: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, and Exponential. By choosing $u$ following this hierarchy, the resulting integral $\int v du$ is generally simpler than the original, as the functions higher in the list simplify significantly when differentiated.

The 'Logarithmic Pattern' recognizes integrals in the form $\int \frac{f'(x)}{f(x)} dx$. On a graph, this represents the area under a reciprocal-style function where the numerator is the exact rate of change of the denominator. The result is always $\ln|f(x)| + C$.

Repeated Integration by Parts is required when the algebraic part of the integrand has a degree higher than 1 (e.g., $x^2 \sin(x)$). In some cases, such as $\int e^x \cos(x) dx$, the process 'loops' and returns to the original integral form, allowing you to solve for the integral algebraically as if it were a variable in an equation.

Visualizing the 'Differential' $du$: In substitution, the term $du = g'(x) dx$ represents the change in the new variable relative to $x$. Visually, if $g(x)$ is steep, a small change in $x$ results in a large change in $u$, and the substitution accounts for this stretching or shrinking of the horizontal units across the area being calculated.