Q: What is the integral of the derivative with respect to x of f divided by the quantity p squared plus q squared f squared with respect to x where f is a function of x and p and q are constants?

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âˆ« f'(x)/( q2f(x)2 - p2) dx = [1/(2pq)ln[(qf(x) - p)/(qf(x) + p)]

âˆ« f'(x)/âˆš(af(x) + b) dx = 2âˆš(af(x) + b)/a + C C is the constant of integration.

âˆ« f'(x)/âˆš[f(x)2 + a] dx = ln[f(x) + âˆš(f(x)2 + a)] + C C is the constant of integration.

Increase in cost: take the first derivative with respect to the unit produced of a cost function. Total cost: sub-in the new quantity into the cost function.

âˆ« f'(x)/(af(x)2 + bf) dx = (1/b)ln[f(x)/(af(x) + b)] + C C is the constant of integration.

Related questions

âˆ« f'(x)/( q2f(x)2 - p2) dx = [1/(2pq)ln[(qf(x) - p)/(qf(x) + p)]

Marginal cost - the derivative of the cost function with respect to quantity. Average cost - the cost function divided by quantity (q).

âˆ« f'(x)/âˆš(af(x) + b) dx = 2âˆš(af(x) + b)/a + C C is the constant of integration.

âˆ« f'(x)/âˆš[f(x)2 + a] dx = ln[f(x) + âˆš(f(x)2 + a)] + C C is the constant of integration.

A variable is a quantity which changes its value through out the program or its lifetime. But a constant is a quantity which does not change its value through out its life time. There are 5 basic constants.

âˆ« f'(x)(af(x) + b)n dx = (af(x) + b)n + 1/[a(n + 1)] + C C is the constant of integration.

The spacial derivative is the measure of a quantity as and how it is being changed in space. This is different from a temporal derivative and partial derivative.

Increase in cost: take the first derivative with respect to the unit produced of a cost function. Total cost: sub-in the new quantity into the cost function.

âˆ« f'(x)/(af(x)2 + bf) dx = (1/b)ln[f(x)/(af(x) + b)] + C C is the constant of integration.

âˆ« [f'(x)g(x) - g'(x)f(x)]/g(x)2 dx = f(x)/g(x) + C C is the constant of integration.

The cost of increasing the production by one unit. Mathematically, this can be derived as the derivative of the total costs with respect to quantity i.e. dc(q)/dq, where c(q) is the cost function and q is quantity.

âˆ« [f'(x)g(x) - g'(x)f(x)]/[f(x)g(x)] dx = ln(f(x)/g(x)) + C C is the constant of integration.