Evaluate the following: 1. F=∫1eln2tdt

Question

Evaluate the following:    1. F=∫1eln2tdt

opsadnojD · Accepted Answer

Step1
The integration properties and methods help to integrate the given function.
Integration by parts formula helps to integrate the given function, which is defined as

\int {(f \cdot g}) d x = f \int g d x - \int {(\frac{d f}{d x} \int g d x}) d x

.
Here f is the first integrable function and g is the second integrable function.
ILATE rule helps to determine the first and second integrable function.
Step2
The given integrand function

I n^{2} t

can be written as the product of 1 by itself.
Take f as the square

\log

function and 1 as the second integrable function.
Here L represents the limits of the integration.
Integrate equation (1) again with respect to t.

\int_{1}^{e} \ln^{2} t d t = \int_{1}^{e} 1 . \ln^{2} t d t

= {{(\ln^{2} t \int 1 d t - \int {(\frac{d (\ln^{2} t)}{d t} \int 1 d t}) d t})}_{1}^{e}

= {{(t \ln^{2} t - \int \frac{2 \ln t}{t} t d t})}_{L}

1)

= {{(t \ln^{2} t - 2 \int \ln t d t})}_{L}

2)

= {{(t \ln^{2} t - 2 (t \ln t - t}))}_{L}

Step 3
Put the limits of the integration into the function present in equation (2) to find the required value of the integration.
Substitute the upper limit into the function and subtract the substitution of the lower limit out of it.

\int_{1}^{e} \ln^{2} t d t = (e \ln^{2} e - 2 (e \ln e - e)) - (1 \ln^{2} 1 - 2 (1 \cdot \ln 1 - 1))

= (e - 2 (e - e)) - (0 - 2 (0 - 1))

= e - 1

= 1.71

Evaluate the following: 1. F=\int_{1}^{e}\ln^{2}t dt

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