Question - Exploration of an Unusual Integral Expression
Solution:
The image displays an integral that needs to be solved. The integral is written as:\[-2 \int_{0}^{t} 0.1t \, dt\]This integral is a bit unusual because the variable of integration, $$ dt $$, matches a variable within the integral's expression, $$ 0.1t $$. Normally, integrals like this one, with a variable as the upper limit of integration and the same variable in the integrand, would be approached using definite integrals. However, this appears to be an error since usually, the integrand should not contain the same variable as the limit of an integral without further context or special instructions like dealing with a parameter.Assuming that there is an error and that the variable in the integrand should be different from the variable of integration, let's proceed with the integral by replacing $$ 0.1t $$ with a constant $$ k $$ (since an integral with respect to $$ dt $$ should not have $$ t $$ as part of the integrand):\[-2 \int_{0}^{t} k \, dt\]Now let's integrate assuming that $$ k $$ is a constant:\[-2k \int_{0}^{t} dt = -2k[t]_{0}^{t} = -2k(t - 0) = -2kt\]If we revert back to using the original variable $$ 0.1t $$, but keep in mind it should be a constant with respect to $$ t $$, we would then shorthand write:\[-2kt\]But since this is against the general rules of calculus and may cause confusion, this solution is provided with caution and under the assumption that the integral's integrand was meant to represent a constant factor, and not the variable of integration $$ t $$.
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