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TASSpecialist MathematicsQuick questions

Unit 4

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

4short 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 uu be the inner function. With u=g(x)u = g(x) and du=gβ€²(x) dxdu = g'(x)\,dx, $∫f(g(x)) gβ€²(x) dx=∫f(u) du. \int f(g(x))\, g'(x)\, dx = \int f(u)\, du. Foradefiniteintegral,changethelimitstovaluesof For a definite integral, change the limits to values of u$ rather than substituting back.
What are integration by parts?
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Integration by parts reverses the product rule: $∫u dvdx dx=uvβˆ’βˆ«v dudx dx. \int u\, \frac{dv}{dx}\, dx = uv - \int v\, \frac{du}{dx}\, dx. Choose Choose utobethefactorthatbecomessimplerwhendifferentiated.Ausefulpriorityforpicking to be the factor that becomes simpler when differentiated. A useful priority for picking u$ 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, $P(x)(xβˆ’a)(xβˆ’b)=Axβˆ’a+Bxβˆ’b, \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)y = f(x) between x=ax = a and x=bx = b is rotated about the xx axis, the solid formed has volume $V=Ο€βˆ«aby2 dx=Ο€βˆ«ab(f(x))2 dx. V = \pi \int_a^b y^2 \, dx = \pi \int_a^b \big(f(x)\big)^2 \, dx. Rotationaboutthe Rotation about the yaxisgives axis gives V = \pi\int_c^d x^2 \, dy$.

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