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Cudi1
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Homework Statement
let u=lnx
du=1/x*dx
dv=cosdx
v=-sin
Homework Equations
Now I am confused as I am getting nowhere with this substiution, i learned the LIPTE rule but its quite confusing, i have a function within a function
The LIPTE rule is a mnemonic device used to remember the steps for integrating by parts. LIPTE stands for: Logarithmic, Inverse trigonometric, Polynomial, Trigonometric, and Exponential. It helps identify which function should be chosen as u and which as dv when using the integration by parts method.
No, the LIPTE rule is not applicable to all integrals. It is most commonly used for integrals involving products of functions, where the product rule cannot be easily applied. It is also useful for integrals involving logarithmic, inverse trigonometric, polynomial, trigonometric, and exponential functions.
To use the LIPTE rule, follow these steps: 1) Identify the functions in the integral that fall under the LIPTE categories. 2) Choose one as u and the other as dv. 3) Take the derivative of u and the antiderivative of dv. 4) Plug these values into the integration by parts formula: ∫udv = uv - ∫vdu. 5) Simplify the resulting integral and solve for the original integral.
Yes, the LIPTE rule can be applied to this integral. In this case, u = ln(x) and dv = cos(x)dx. The derivative of u is 1/x and the antiderivative of dv is sin(x). Plugging these values into the integration by parts formula results in ∫cos(lnx)dx = ln(x)sin(x) - ∫(1/x)sin(x)dx. This integral can then be solved using the substitution method.
Yes, there are other methods for integrating this function, such as using the power rule or substitution method. However, the LIPTE rule is often the most efficient and straightforward method for this specific integral.