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MATH 54 CHAPTER 1
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- 1THE IMAGE SHOWS WHICH TECHNIQUE OF INTEGRATION?INTEGRATION BY PARTS
- 2THE INTEGRATION BY PARTS FORMULATHE INTEGRAL OF U D V EQUALS U V MINUS THE INTEGRAL OF V D U
- 3PROVIDE THE PROOF FOR FORMULA ✯
- 4RANDOM NOTES #1When applying Formula ✯ to an integral, we begin by letting one part of the integrated correspond to dv. The expression we choose for dv must include the differential dx. After selecting dv, we designate the remaining part of the integrand u and then find du. Since this process involves splitting the integrand into two parts, the use of ✯ is referred to as Integrating by Parts. A proper choice for dv is crucial. We usually let dv equal the most complicated part of the integrand that can be readily integrated. The following examples illustrate this method of integration.
- 5\int xe^{2x}\;\mathrm{d}x\int xe^{2x}\;\mathrm{d}x = \frac{1}{2}xe^{2x}-\frac{1}{4}e^{2x} + C
- 6\int x\sec^2{x}\;\mathrm{d}x\int x\sec^2{x}\;\mathrm{d}x = x\tan{x} + \ln{|\cos{x}|}
- 7\int \ln{x}\;\mathrm{d}x\int \ln{x}\;\mathrm{d}x = x\ln{x} - x + C
- 8\int x^2\ln{x}\;\mathrm{d}x\int x^2\ln{x}\;\mathrm{d}x = \frac{x^3}{3} \ln{x} - \frac{x^3}{9} + C
- 9What does LIATE stand for?The order in which we prioritize the "let u" part of the ✯.
L - Logarithmic Function
I - Inverse Trigonometric
A - Algebraic Expression (Polynomials included)
T - Trigonometric Functions
E - Exponential Function - 10\int e^x \cos{x}\;\mathrm{d}x\int e^x \cos{x}\;\mathrm{d}x = \frac{1}{2}e^x(\sin{x}+\cos{x}) + C
- 11\int \tan^{-1}{x}\;\mathrm{d}x\int \tan^{-1}{x}\;\mathrm{d}x = x\tan^{-1}{x} - \frac{1}{2}\ln{|1+x^2|} + C
- 12\int e^x\sin{x}\;\mathrm{d}x\int e^x\sin{x}\;\mathrm{d}x = \frac{e^x\sin{x}}{2} - \frac{e^x\cos{x}}{2} + C
- 13\int x^2e^{3x}\;\mathrm{d}x\int x^2e^{3x}\;\mathrm{d}x = \frac{2xe^{3x}}{3} - \frac{2xe^{3x}}{9} + \frac{2xe^{3x}}{27} + C
- 14\int x^3\cos{x}\;\mathrm{d}x\int x^3\cos{x}\;\mathrm{d}x = x^3\sin{x} + 3x^2\sin{x} - 6x\sin{x} - 6cos{x} + C
- 15\int_{1}^{e} \ln{x}\;\mathrm{d}x\int_{1}^{e} \ln{x}\;\mathrm{d}x = 1
- 16\int x^2\cos{x}\;\mathrm{d}x\int x^2\cos{x}\;\mathrm{d}x = x^2\sin{x} + 2x\cos{x} - 2\sin{x} + C
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- Mathematics Fichas sobre MATH 54 CHAPTER 1, creado por Jarell Paras el 02/03/2021.
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