적분 전체 문제

For the following exercises, evaluate the integral.

$\int \frac{3{x}^{2}+2}{{x}^{2}}dx$

For the following exercises, evaluate the integral.

$\int (\sec x\tan x+4x)dx$

For the following exercises, evaluate the integral.

$\int \left(4\sqrt{x}+\sqrt[4]{x}\right)dx$

For the following exercises, evaluate the integral.

$\int ({x}^{-1/3}-{x}^{2/3})dx$

For the following exercises, evaluate the integral.

$\int \frac{14{x}^{3}+2x+1}{{x}^{3}}dx$

For the following exercises, evaluate the integral.

$\int ({e}^{x}+{e}^{-x})dx$

For the following exercises, solve the initial value problem.

${f}^{'}(x)={x}^{-3},f(1)=1$

For the following exercises, solve the initial value problem.

${f}^{'}(x)=\sqrt{x}+{x}^{2},f(0)=2$

For the following exercises, solve the initial value problem.

${f}^{'}(x)=\cos x+{\sec }^{2}(x),f\left(\frac{\pi }{4}\right)=2+\frac{\sqrt{2}}{2}$

For the following exercises, solve the initial value problem.

${f}^{'}(x)={x}^{3}-8{x}^{2}+16x+1,f(0)=0$

For the following exercises, solve the initial value problem.

${f}^{'}(x)=\frac{2}{{x}^{2}}-\frac{{x}^{2}}{2},f(1)=0$

For the following exercises, find two possible functions $f$ given the second- or third-order derivatives.

${f}^{''}(x)={x}^{2}+2$

For the following exercises, find two possible functions $f$ given the second- or third-order derivatives.

${f}^{''}(x)=1+x$

For the following exercises, find two possible functions $f$ given the second- or third-order derivatives.

$f'''(x)=\cos x$

For the following exercises, find two possible functions $f$ given the second- or third-order derivatives.

$f'''(x)=8{e}^{-2x}-\sin x$

A car is being driven at a rate of $40$ mph when the brakes are applied. The car decelerates at a constant rate of $10$ ft/sec². How long before the car stops?

In the preceding problem, calculate how far the car travels in the time it takes to stop.

You are merging onto the freeway, accelerating from rest at a constant rate of $12$ ft/sec². How long does it take you to reach merging speed at $60$ mph?

Based on the previous problem, how far does the car travel to reach merging speed?

A car company wants to ensure its newest model can stop in $8$ sec when traveling at $75$ mph. If we assume constant deceleration, find the value of deceleration that accomplishes this.