Integral calculus and economic applications

Compute $\int \left(4x^3-6\sqrt{x}+\frac{2}{x}\right)dx$ for $x>0$.

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Compute $\int x^2{\left(x^3+1\right)}^{4}dx$.

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Compute $\int x\sqrt{2x+1}dx$ with the substitution $t=2x+1$.

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Compute $\int x\ln xdx$ for $x>0$.

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Compute $\int \frac{x^2+1}{x-1}dx$.

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Compute $\int \frac{x+7}{x^2+2x-3}dx$.

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Compute ${\int }_0^4\frac{1}{1+\sqrt{x}}dx$.

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The price of a good in the first $4$ months of sales is $p(t)=20+3\sqrt{t}$ euros, with $t$ in months. Compute the average price over the interval $[0,4]$.

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Let $F(x)={\int }_0^x\left(4-t^2\right)dt$. Find $F^{\prime }(x)$ and the relative maximum and minimum points of $F$, and compute $F(2)$.

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Compute the area of the region bounded by the parabola $y=x^2$ and by the line $y=6-x$.

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Compute ${\int }_0^{+\infty }x{e}^{-2x}dx$.

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The marginal cost of a firm is $C^{\prime }(q)=3q^2-24q+60$ and the fixed cost is $200$ euros. Find the total cost $C(q)$ and the additional cost of raising production from $4$ to $10$ units.

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The inverse demand of a good is $p_d(q)=52-q^2$ and the inverse supply is $p_s(q)=12+6q$, with $q\ge 0$. Find the equilibrium and the consumer and producer surpluses.

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An investment produces a continuous cash flow of $5000$ euros per year. With instantaneous intensity of interest $\delta =0.05$, compute the present value of the flow if it lasts $4$ years and if it has no end.

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