Antiderivatives: Substitution

Find the following indefinite integral using the method of substitution.\[\int 4xe^{-5x^2+1} \ dx\]

Find the following indefinite integral.\[\int \frac{\ln(x)}{x} \ dx\]

Find the following indefinite integral.\[\int \frac{e^{2x}}{e^{2x}+4} \ dx\]

Find the following indefinite integral.\[\int \frac{2x^4}{\sqrt[3]{8x^5-9}} \ dx\]

Find \(f(x)\) using the given information.\[f'(x)=\frac{\ln(x)}{x}+4x^3-12x \quad\text{ and } \]\[f(e)=e^4-6e^2+82\]

Dice Dice Baby has found that its marginal profit function for making and selling polyhedral dice sets is given by \(P'(x)=\dfrac{6x}{\sqrt{7x^2+20}}\) dollars per dice set when \(x\) dice sets are made and sold. If the company's profit from making and selling 500 dice sets is \(\$240,\) what is the company's profit from making and selling 600 dice sets?