Deck 30: Integration

Question 1

Find the indefinite integral: $\int 4 d x$

Answer 1

$4 x+C$

Question 2

Find the indefinite integral: $\int \sqrt[5]{x} \cdot d x$

Answer 2

$\frac{5}{6} \sqrt[5]{x^{6}}+C$

Question 3

Find the indefinite integral: $\int 18 x^{5} d x$

Answer 3

$3 x^{6}+C$

Question 4

Find the indefinite integral: $\int(x+\sqrt{x})^{2} d x$

Question 5

Find the indefinite integral: $\int 5 \sqrt{x}\left(x^{2}-1\right) d x$

Question 6

Find the function that has the derivative: $\frac{d y}{d x}=\frac{4}{x^{6}}$

Question 7

Find the function that has the derivative: $y^{\prime}=2 x^{3}\left(3 x^{4}-5\right)$

Question 8

Find the function whose derivative is given: $\frac{d y}{d x}=5 x^{4}+\frac{2}{\sqrt[3]{x}}+1$

Question 9

Integrate: $\int\left(8 x^{3}+5 x-1\right) d x$

Question 10

Integrate: $\int \sqrt{x}(1-x) d x$

Question 11

Find the function that has the derivative $\frac{d y}{d x}=9 x^{2}-2 x^{3}$ .

Question 12

Find the indefinite integral: $\int(\sqrt{2}-2 x) d x$

Question 13

Find the indefinite integral: $\int \frac{x^{2}+3 x+2}{x+1} \cdot d x$

Question 14

Integrate: $\int\left(x^{3}-4 x+5\right) d x$

Question 15

Integrate: $\int \frac{4 x-1}{\sqrt{x}} d x$

Question 16

Find the function that has the derivative $\frac{d y}{d x}=2 x(x+5)$ .

Question 17

Integrate: $\int\left(x^{3}-2 x\right)^{3}\left(6 x^{2}-4\right) d x$

Question 18

Integrate: $\int \frac{d x}{\sqrt{x-1}}$

Question 19

Integrate: $\int \frac{4 x^{2} d x}{1-x^{3}}$

Question 20

Integrate: $\int \frac{x+3}{x+1} \cdot d x$

Question 21

Find the function whose derivative is given:
$y=\ln |x|+\frac{3}{2} x^{2}+C$

Question 22

Integrate: $\int 5\left(x^{5}-10 x\right)^{2}\left(x^{4}-2\right) d x$

Question 23

Integrate: $\int \frac{z d z}{\sqrt{z^{2}+1}}$

Question 24

Integrate: $\int \frac{d w}{2 w+5}$

Question 25

Integrate: $\int \frac{3 x^{3}+8 x^{2}+4 x-1}{x} \cdot d x$

Question 26

Integrate: $\int \frac{x^{2}-2}{x} d x$

Question 27

Integrate: $\int \frac{40 x+8}{\sqrt{5 x^{2}+2 x}} d x$

Question 28

Integrate: $\int 32 x^{3}\left(3-2 x^{4}\right)^{6} d x$

Question 29

Integrate: $\int \frac{5}{2 x} d x$

Question 30

Find the function whose derivative is given: $\frac{d y}{d x}=x\left(x^{2}+2\right)^{4}$

Question 31

Integrate: $\int \frac{3 x-2}{3 x^{2}-4 x} d x$

Question 32

Integrate: $\int \frac{y d y}{y^{2}+1}$

Question 33

Find the function that has the derivative $\frac{d y}{d x}=4 x$ and graph this function for $\mathrm{C}=-1, \mathrm{C}=0$ , and $\mathrm{C}=1$ .

Question 34

Write the function with the given derivative that passes through the given points: $y^{\prime}=\frac{1}{\sqrt{3 x}}$ . Passes through $(3,3)$ .

Question 35

Find the equation of the curve through $(-1,-5)$ that has $y^{\prime \prime}=2$ and $y^{\prime}(-1)=2$ .

Question 36

Find the equation of the curve that passes through the point $(4,-5)$ , has slope -130 at the point, and has the second derivative $y^{\prime \prime}=\frac{1}{2} \sqrt{x}-3 x^{2}$ .

Question 37

Write the function that has the derivative $y^{\prime}=\frac{3}{2} \sqrt{x}$ and passes through the point $(1,3)$ .

Question 38

If $\frac{d y}{d x}=6 x\left(x^{2}+2\right)^{2}$ , and $y=25$ when $x=1$ , find the value of $y$ when $x=-1$ .

Question 39

If $y^{\prime \prime}=\sqrt{x}$ , find the equation of a curve that passes through the point $(0,0)$ , whose tangent at that point is perpendicular to the line $4 x+y=8$ .

Question 40

Find the function that goes through the point $(4,5)$ and has the derivative $\frac{d y}{d x}=8 x-\frac{4}{\sqrt{x}}+2$ .

Question 41

Find the function that goes through the point $(2,-7)$ and has the derivative $\frac{d y}{d x}=(12 x-15)\left(2 x^{2}-5 x\right)^{5}$ .

Question 42

If $\frac{d^{2} y}{d x^{2}}=24(2 x-1)^{2}$ , find the equation of a curve that passes through the point $(1,3)$ and whose tangent at that point has a slope of 2 .

Question 43

Find the function that goes through the point $(-1,3)$ and has the derivative $\frac{d y}{d x}=\frac{8}{2 x+1}$ .

Question 44

Evaluate: $\int_{1}^{4} 3 x^{5} d x$

Question 45

Evaluate the definite integral: $\int_{0}^{8} \frac{d x}{\sqrt{x+1}}$

Question 46

Evaluate the definite integral to three significant digits: $\int_{2}^{5} \frac{x d x}{x^{2}+1}$

Question 47

Evaluate the definite integral to the nearest whole number: $\int_{0}^{10}\left(x^{2}-x^{3}\right) d x$

Question 48

Evaluate the definite integral to four significant digits: $\int_{2}^{3} \frac{x^{2}}{x^{3}-1} \cdot d x$

Question 49

Integrate: $\int_{1}^{5} \frac{4 d x}{1-2 x}$ . Express your answer to two decimal places.

Question 50

Integrate: $\int_{0}^{15}\left(8 x^{3}-12 x^{2}+5 x\right) d x$

Question 51

Integrate: $\int_{2}^{6} \sqrt[3]{x^{5}} d x$

Question 52

Integrate: $\int_{0}^{1}(6 x+9)\left(x^{2}+3 x\right)^{3} d x$

Question 53

Integrate: $\int_{1}^{4} \frac{x^{2}-1}{x} d x$

Question 54

Evaluate the expression: $\sum_{n=1}^{4} n(n+1)$

Question 55

Find the approximate area (in square units) under the curve by the midpoint method, using panels 2 units wide: $y=x^{3}+1$ from $x=0$ to $x=6$

Question 56

Evaluate: $\sum_{n=1}^{6}\left(n^{2}-n\right)$

Question 57

Find the approximate area (in square units) under the curve by the midpoint method, using panels 2 units wide: $y=4 \sqrt{x}$ from $x=0$ to $x=10$

Question 58

Find the approximate area (in square units) under the curve by the midpoint method, using panels 1 units wide: $y=x^{2}$ from $x=0$ to $x=4$

Question 59

Find the area between the curve, the given lines, and $x$ axis: $y=\frac{1}{\sqrt{2 x}}$ from $x=2$ to $x=8$

Question 60

Find the area between the curve, the given lines, and $x$ axis: $y=2 x+2 \sqrt{x}$ from $x=1$ to $x=7$

Question 61

Find the area bounded by the curve, the given lines, and the $x$ axis: $y=4-x^{2}$ from $x=-1$ to $x=2$ .

Question 62

Find the area in the first quadrant that is bounded by the function $y=4+4 x-3 x^{2}$ .

Question 63

Find the area (in square units) bounded by the curve, the given lines, and the $x$ -axis: $y=\frac{1}{x^{2}}$ from $x=1$ to $x=5$

Question 64

Find the area (in square units, to three significant digits) bounded by the curve, the given lines, and the $x$ -axis: $y=\frac{x+1}{x+2}$ from $x=2$ to $x=4$

Question 65

Find the area (in square units) bounded by the curve $y=10+3 x-x^{2}$ and the $x$ -axis.

Question 66

Find the area (in square units) bounded by the curve $y=\frac{2}{3 x-5}$ , the x-axis, $x=2$ , and $x=8$ .

Question 67

Find the area (in square units) bounded by the curve $y=\frac{4}{\sqrt{x+1}}$ , the x-axis, $x=0$ , and $x=8$ .

Question 68

Find the area (in square units) bounded by the curve $y=\sqrt{x}(2-x)$ and the $x$ -axis.

Question 69

Find the area (in square units, to three significant digits) bounded by the curve $y=25-x^{2}$ and the $x$ -axis.