Week 15 - Techniques of Integration #

Integration by Parts #

• Every differentiation rule has a corresponding integration rule.
• The rule that corresponds to the Product Rule for differentiation is called the rule for integration by parts.
• The Product Rule states that if f and g are differentiable functions, then $$\frac{d}{dx}[f(x)g(x)] = f(x)g’(x) + f’(x)g(x)$$
• In the notation for indefinite integrals this equation becomes $$\int [f(x)g’(x) + f’(x)g(x)] dx = f(x)g(x)$$
• We can rearrange this equation as $$\int f(x)g’(x) dx = f(x)g(x) - \int f’(x)g(x)$$
• Let $u = f(x)$ and $v = g(x)$, we can simplify the equation: $$\int u\\ dv = uv - \int v\\ du$$

Examples #

1. $\int x e^x$
   • Let $u = x$ and $dv = e^x dx$
   • $\int x e^x = x e^x - \int x \cdot e^x = x e^x - e^x + C$
2. $\int \log{x} dx$
   • Let $u = \log{x}$ and $dv = dx$
   • $\int \log{x} dx = x\log{x} - \int \frac{d}{dx}\log{x} \cdot x dx = x\log{x} - \int 1\\ dx = x\log{x} - x$
3. $\int e^x \cos{x} dx$
   • Let $u = e^x$ and $dv = \cos{x}\\ dx$
   • $\int \cos{x} e^x\\ dx = e^x \cdot \sin{x} - \int e^x \cdot \sin{x}\\ dx$
   • Let’s calculate the part $\int e^x \cdot \sin{x}\\ dx$
     • Let $u = e^x$ and $dv = \sin{x}\\ dx$
     • $\int e^x \cdot \sin{x}\\ dx = e^x \cdot (-\cos{x}) - \int e^x \cdot (-\cos{x})\\ dx$
     • $\int e^x \cdot \sin{x}\\ dx = - e^x \cdot \cos{x} + \int e^x \cdot \cos{x}\\ dx$
   • Then we get $\int e^x \cos{x}\\ dx = e^x \cdot \sin{x} + e^x \cdot \cos{x} - \int e^x \cdot \cos{x}\\ dx$
   • So, $\int e^x \cos{x}\\ dx = \frac{e^x \cdot \sin{x} + e^x \cdot \cos{x} }{2}$

When to Use Parts? #

• Example $\int e^{\sqrt{x} } dx$
• First we need to use Substitution rule to replace $\sqrt{x}$
• Let’s set $u = \sqrt{x}$, then $x = u^2, dx = 2u$
• We rewrite the function $\int e^u \cdot 2u\\ du$
• Now we can use Parts:
  • Let $v = 2u, dw = e^u\\ du$, then
  • $\int e^u \cdot 2u\\ du = 2u \cdot e^u - \int 2e^u du = 2u \cdot e^u - 2e^u + C = 2e^{\sqrt{x} }(\sqrt{x}-1) + C$

Integrate Powers of Sines and Cosines #

• If the power of cosine or sine is odd:
  • Example $\int \sin^{(2n+1)}{x} \cdot \cos^{2n}{x}\\ dx$, ($n \in \mathbb{R}$)
  • We can’t use the Substitution rule directly, because whichever we choose to be $u$ (sine or cosine), we will always get a part that can’t deal with.
    • Like, if we choose $u = \sin{x}, du = \cos{x}\\ dx$, then the part $\cos^{2n}{x}$ will not be able to transfer.
  • Instead of making substitution immediately, we can trade a pair of sines for a pair of cosines.
    • $= \int \sin{x} \cdot \sin^{2n}{x} \cdot \cos^{2n}{x} \\ du$
    • $= \int \sin{x} \cdot (1-\cos^{2}{x})^n \cdot \cos^{2n}{x} \\ du$
  • Set $u = \cos{x}, du = -\sin{x}\\ dx$, we get:
    • $= - \int (1-u^{2})^n \cdot u^{2n} \\ du$
    • $= - \int u^{2n} - u^{4n} \\ du$
    • $= - \frac{1}{2n+1}u^{2n+1} + \frac{1}{4n+1}u^{4n} + C$
    • $= - \frac{1}{2n+1}\cos^{2n+1}{x} + \frac{1}{4n+1}\cos^{4n}{x} + C$
  • The conclusion is, we can always transfer the odd one to $\sin^{2n}{x} \cdot \sin{x}$ or $\cos^{2n}{x} \cdot \cos{x}$, and use the equation $\sin^2{x} + \cos^2{x} = 1$ to transfer between $\sin$ and $\cos$ .
• If the powers of both sine and cosine are even, use the half-­angle identities:$$ \sin^2{x} = \frac{1}{2}(1 - \cos(2x)),\\ \cos^2{x} = \frac{1}{2}(1 + \cos(2x)) $$
  • It is sometimes helpful to use the identity $$\sin{x}\cos{x} = \frac{1}{2}\sin{2x}$$
  • For example:
    • $$\begin{aligned} \int_0^{\pi} \sin^2{x} \\ dx &= \frac{1}{2} \int_0^{\pi}(1 - \cos{2x})\\ dx \\ &= \big[\frac{1}{2}(x-\frac{1}{2}\sin{2x})\big]_0^{\pi} \\ &= \frac{1}{2}(\pi - \frac{1}{2} \sin{2\pi}) - \frac{1}{2}(0 - \frac{1}{2}\sin{0}) = \frac{1}{2}\pi \end{aligned}$$

Tables of Indefinite Integrals #

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  • $tan^{-1}x = \arctan{x}$

  • Hyperbolic functions *In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. –from Wikipedia

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Words #

• vice versa [,vaisi’və:sə] 反之亦然