Deck 5: The Integral

Question 1

Integrate $\int\left(6 x^{2}-4 x+9\right) d x$ .

A) $12 x-4$
B) $2 x^{3}-2 x^{2}+C$
C) $2 x^{3}-2 x^{2}+9 x+C$
D) $2 x^{3}-x^{4}+9 x+C$

Answer 1

$2 x^{3}-2 x^{2}+9 x+C$

Question 2

Integrate $\int \frac{2 x}{\sqrt{x^{2}+3}} d x$ .

A) $\frac{1}{2} \sqrt{\mathrm{x}^{2}+3}+\mathrm{C}$
B) $2 \sqrt{x^{2}+3}+C$
C) $\sqrt{x^{2}+3}+C$
D) $\frac{4}{3}\left(x^{2}+3\right)^{3 / 2}+C$

Answer 2

$2 \sqrt{x^{2}+3}+C$

Question 3

Integrate $\int \frac{1}{(5 x+2)^{2}} d x$ .

A) $\ln (5 x+2)^{2}+C$
B) $\frac{-1}{5(5 x+2)}+C$
C) $\frac{-1}{5 x+2}+C$
D) $\frac{1}{5} \ln (5 x+2)^{2}+C$

Answer 3

$\frac{-1}{5(5 x+2)}+C$

Question 4

Recall that the voltage across a capacitor is given by $V_{C}=\frac{1}{C} \int i(t) d t$ . Find the formula for $V_{C}$ if $C=$ .01 farad, $i(t)=2 t+3$ amperes, and $V_{C}(0)=12$ volts.

A) $V_{C}(t)=100 t^{2}+300 t^{2}+1200$ volts
B) $V_{C}(t)=t^{2}+3 t+1200$ volts
C) $V_{C}(t)=100 t^{2}+300 t+12$ volts
D) $V_{C}(t)=t^{2}+3 t+12$ volts

Question 5

Find the area under the curve $y=4-3 x^{2}$ from $x=0$ to $x=1$ .

A) 10
B) 4
C) 3
D) 7

Question 6

Evaluate the definite integral $\int_{-2}^{0}\left(2 x^{3}-4 x\right) d x$ .

A) 0
B) -2
C) 4
D) 2

Question 7

Evaluate the definite integral $\int_{0}^{1}(2 x+1)^{3} d x$ .

A) 20
B) 10
C) $\frac{81}{4}=20.25$
D) 40

Question 8

For the problems below, integrate.
-8 $\int 3 x^{23} d x$

Question 9

For the problems below, integrate.
-9 $\int(3 x+1)^{1 / 2} d x$

Question 10

For the problems below, integrate.
-10 $\int \frac{x^{2} d x}{\left(4 x^{3}-5\right)^{3}}$

Question 11

For the problems below, integrate.
-11 $\int \frac{\left(x^{4}+2 x^{2}\right) d x}{\sqrt[4]{3 x^{5}+10 x^{3}}}$

Question 12

For the problems below, integrate.
-Find the equation of the curve $y=f(x)$ satisfying the conditions: $\frac{d x}{d y}=\left(x^{2}-6\right)^{3} x d x$ and passing through $(2,5)$ .

Question 13

For the problems below, integrate.
-A ball is hurled straight up from the top of a $28 \mathrm{ft}$ building at a velocity of $44 \mathrm{ft} / \mathrm{s}$ . a) Find the equation describing the altitude of the ball from the ground. b) How long does it take for the ball to hit the ground? Round to three significant digits.

Question 14

For the problems below, integrate.

-A capacitor with a capacitance $10^{-3} \mathrm{~F}$ has a voltage of $120 \mathrm{~V}$ across it. At a given instant (t $=0$ ) the capacitor is connected to a source that sends a current $i=6 \sqrt{t}+.01 \mathrm{~A}$ through the circuit. Find the voltage across the capacitor when $\mathrm{t}=0.36 \mathrm{~s}$ .

Question 15

For the problems below, find the area under the curve.

- $y=-x^{2}+3 x+10$ from $x=1$ to $x=4$

Question 16

For the problems below, find the area under the curve.

- $y=\sqrt{4 x+1}$ from $x=2$ to $x=6$

Question 17

For the problems below, find the area under the curve.

-Find the area bounded by the $x$ - axis and $y=-x^{2}+10 x-16$ .

Question 18

For the problems below, evaluate each definite integral.

- $\int_{2}^{3}\left(6 x^{2}-x-6\right) d x$

Question 19

$\int_{-3}^{0} 2 x \sqrt{25-x^{2}} d x$

Question 20

$\int_{0}^{4} \frac{18 x}{\sqrt{9 x^{2}+25}} d x$

Question 21

$\int_{2}^{5}\left(x^{3}-9 x^{2}+6 x-3\right) d x$