§-Quick questions

TASSpecialist MathematicsUnit 4

Quick questions on Integration techniques and applications - TCE Mathematics Specialised (Tasmania)

5short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is substitution?

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Substitution reverses the chain rule. If part of the integrand is the derivative of another part, let

u

be the inner function. With

u = g(x)

and

du = g'(x)\,dx

, $

\int f(g(x))\, g'(x)\, dx = \int f(u)\, du.

For a definite integral, change the limits to values of

u$ rather than substituting back.

What are integration by parts?

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Integration by parts reverses the product rule: $

\int u\, \frac{dv}{dx}\, dx = uv - \int v\, \frac{du}{dx}\, dx.

Choose

to be the factor that becomes simpler when differentiated. A useful priority for picking

u$ is logarithms, then inverse trigonometric functions, then algebraic, then trigonometric, then exponential.

What are partial fractions?

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A proper rational function with a factorised denominator can be split into simpler fractions. For distinct linear factors, $

\frac{P(x)}{(x - a)(x - b)} = \frac{A}{x - a} + \frac{B}{x - b},

$ and each piece integrates to a logarithm.

What is volumes of revolution?

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When the region under

y = f(x)

between

x = a

and

x = b

is rotated about the

x

axis, the solid formed has volume $

V = \pi \int_a^b y^2 \, dx = \pi \int_a^b \big(f(x)\big)^2 \, dx.

Rotation about the

axis gives

V = \pi\int_c^d x^2 \, dy$.

What are integration using trigonometric identities?

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Powers of

\sin

and

\cos

are integrated by first reducing the power with an identity. For even powers use the double-angle forms

\sin^2\theta = \dfrac{1 - \cos 2\theta}{2}

and

\cos^2\theta = \dfrac{1 + \cos 2\theta}{2}

; for odd powers peel off one factor and convert the rest with

\sin^2\theta + \cos^2\theta = 1

, then substitute.

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