적분 전체 문제

Evaluate the following integrals.

${\int }_{0}^{4}\frac{3t}{\sqrt{1+6{t}^{2}}}dt$

Evaluate the following integrals.

${\int }_{\pi /3}^{\pi /2}2\sec (2\theta )\tan (2\theta )d\theta$

Evaluate the following integrals.

${\int }_{0}^{\pi /4}{e}^{{\cos }^{2}x}\sin x\cos xdx$

Find the antiderivative.

$\int \frac{dx}{{(x+4)}^{3}}$

Find the antiderivative.

$\int x\ln ({x}^{2})dx$

Find the antiderivative.

$\int \frac{4{x}^{2}}{\sqrt{1-{x}^{6}}}dx$

Find the antiderivative.

$\int \frac{{e}^{2x}}{1+{e}^{4x}}dx$

Find the derivative.

$\frac{d}{dt}{\int }_{0}^{t}\frac{\sin x}{\sqrt{1+{x}^{2}}}dx$

Find the derivative.

$\frac{d}{dx}{\int }_{1}^{{x}^{3}}\sqrt{4-{t}^{2}}dt$

Find the derivative.

$\frac{d}{dx}{\int }_{1}^{\ln (x)}(4t+{e}^{t})dt$

Find the derivative.

$\frac{d}{dx}{\int }_{0}^{\cos x}{e}^{{t}^{2}}dt$

If the average cost per gigabyte of RAM in 2010 is $12, find the average cost per gigabyte of RAM in 1980.

The average cost per gigabyte of RAM can be approximated by the function $C(t)=8,500,000{(0.65)}^{t},$ where $t$ is measured in years since 1980, and $C$ is cost in US$. Find the average cost per gigabyte of RAM for 1980 to 2010.

Find the average cost of 1GB RAM for 2005 to 2010.

The velocity of a bullet from a rifle can be approximated by $v(t)=6400{t}^{2}-6505t+2686,$ where $t$ is seconds after the shot and $v$ is the velocity measured in feet per second. This equation only models the velocity for the first half-second after the shot: $0\le t\le 0.5.$ What is the total distance the bullet travels in 0.5 sec?

What is the average velocity of the bullet for the first half-second?

Define the term “antiderivative” in your own words.

Is it more accurate to refer to “the” antiderivative of $f(x)$ or “an” antiderivative of $f(x)$?

Use your own words to define the indefinite integral of $f(x)$.

Fill in the blanks: “Inverse operations do the _____ things in the _____ order.”