E078 Section 4 — Techniques of Integration

Calculus 2 for Engineers (WGU E078), Section 4: integration by parts, trig integrals, trig substitution, partial fractions, numerical integration, improper integrals, Laplace transform. Built from the zyBooks Section 4 study materials; not official WGU material.

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Integration by parts formula

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Integration by parts for definite integrals

3

Where does integration by parts come from?

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LIATE (choosing u in parts)

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Why are $\ln x$ and inverse trig chosen as $u$?

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$\int x\sin x\,dx = \,?$

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$\int \ln x\,dx = \,?$

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$\int \tan^{-1}x\,dx = \,?$

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When do you apply integration by parts twice?

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The 'loop' trick in integration by parts

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$\int e^x\sin x\,dx = \,?$

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Do you need $+K$ on $v$ in integration by parts?

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Reduction formula: $\int x^n e^x\,dx$

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$\int \cos^j x\sin^k x\,dx$ with $k$ (sine power) odd

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$\int \cos^j x\sin^k x\,dx$ with $j$ (cosine power) odd

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$\int \cos^j x\sin^k x\,dx$ with both powers even

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Power-reducing identity for $\sin^2 x$

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Power-reducing identity for $\cos^2 x$

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$\int \sin^2 x\,dx = \,?$

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$\int \cos^3 x\,dx = \,?$

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$\sin(ax)\sin(bx) = \,?$

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$\sin(ax)\cos(bx) = \,?$

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$\cos(ax)\cos(bx) = \,?$

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$\int \tan x\,dx = \,?$

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$\int \sec x\,dx = \,?$

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$\int \tan^k x\sec^j x\,dx$ with $j$ (sec power) even

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$\int \tan^k x\sec^j x\,dx$ with $k$ (tan power) odd

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$\int \tan^k x\,dx$ with $k$ odd and no sec

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$\int \sec^3 x\,dx = \,?$

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Reduction formula for $\int \sec^n x\,dx$

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Reduction formula for $\int \tan^n x\,dx$

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Trig substitution for $\sqrt{a^2 - x^2}$

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Trig substitution for $\sqrt{a^2 + x^2}$

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Trig substitution for $\sqrt{x^2 - a^2}$

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Reference triangle for $x = a\sin\theta$

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Reference triangle for $x = a\tan\theta$

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Reference triangle for $x = a\sec\theta$

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First step before any trig substitution?

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$\int \frac{dx}{\sqrt{1 + x^2}} = \,?$

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Trig sub with a definite integral: what about the limits?

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When must you do long division before partial fractions?

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Partial fraction term for a nonrepeated linear factor $(ax + b)$

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Partial fraction terms for a repeated linear factor $(ax + b)^n$

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Partial fraction term for an irreducible quadratic $ax^2 + bx + c$

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When is a quadratic $ax^2 + bx + c$ irreducible?

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Method of strategic substitution

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Method of equating coefficients

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$\int \frac{du}{u^2 + a^2} = \,?$

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How do you integrate $\frac{Ax + B}{x^2 + bx + c}$ with an irreducible quadratic?

50

Partial fractions on a non-rational integral, e.g. $\int \frac{\cos x}{\sin^2 x - \sin x}\,dx$

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$\Delta x$ for the numerical integration rules

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Midpoint rule $M_n$

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Trapezoidal rule $T_n$

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Simpson's rule $S_n$

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What must be true about $n$ for Simpson's rule?

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What does Simpson's rule approximate $f$ with?

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$T_n$ in terms of left and right sums

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$S_{2n}$ in terms of $M_n$ and $T_n$

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$f$ concave up: does $T_n$ over- or underestimate?

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Absolute error

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Relative error

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Error bound for the midpoint rule

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Error bound for the trapezoidal rule

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Error bound for Simpson's rule

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Finding $n$ from an error bound: what do you do with a decimal answer?

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Skill Shop (4.6) takeaway: midpoint vs. trapezoid vs. Simpson

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Definition: $\int_a^{+\infty} f(x)\,dx$

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Definition: $\int_{-\infty}^{+\infty} f(x)\,dx$

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Improper integral with $f$ discontinuous at $b$

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Improper integral with an infinite discontinuity at $c$ inside $(a, b)$

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$\int_1^{+\infty} \frac{1}{x}\,dx$

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$\int_1^{+\infty} \frac{1}{x^p}\,dx$

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Comparison theorem ($0 \le f \le g$ on $[a, +\infty)$)

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Gabriel's Horn ($y = 1/x$, $x \ge 1$)

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Why is $\int_{-1}^{1} \frac{dx}{x^2}$ not equal to $-2$?

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Laplace transform definition

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Laplace transforms of $1$ and $t$

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Laplace transform of a derivative