Integrate using substitution, partial fractions and trig identities, and apply integration to volumes and differential equations

What this dot point is asking

SCSA Unit 4 brings together a toolkit of integration methods and their applications. You choose the technique from the structure of the integrand, evaluate definite integrals, find volumes generated by rotating a region about an axis, and solve first-order separable differential equations including growth, decay and logistic models.

Substitution

Reverse the chain rule. Choose $u = g(x)$ so that $du = g'(x)\,dx$ appears (up to a constant) in the integrand, rewrite everything in terms of $u$, integrate, and for a definite integral either change the limits to $u$-values or convert back to $x$.

Partial fractions

A proper rational function with a factorable denominator splits into simpler fractions. For distinct linear factors,

and each piece integrates to a logarithm. If the numerator degree is greater than or equal to the denominator degree, perform polynomial division first.

Trigonometric integrals

Use identities to rewrite integrands into directly integrable forms. The key ones are the double-angle identities for $\cos^2$ and $\sin^2$:

For example $\displaystyle\int \cos^2 x\,dx = \int \frac{1 + \cos 2x}{2}\,dx = \frac{x}{2} + \frac{\sin 2x}{4} + C$.

Volumes of revolution

Rotating the region under $y = f(x)$ between $x = a$ and $x = b$ about the $x$-axis generates a solid of volume

Rotation about the $y$-axis uses $V = \pi\displaystyle\int_c^d x^2\,dy$ with $x$ expressed in terms of $y$.

Differential equations

A separable equation $\dfrac{dy}{dx} = f(x)g(y)$ is solved by separating variables and integrating both sides:

This produces a general solution with one constant; an initial condition fixes the constant. Exponential growth and decay ($\frac{dy}{dt} = ky$) and logistic models appear in context.