Chapter 5

Use the Substitution Formula in Theorem 7 to evaluate the integrals in Exercises \(1-46\). a. \(\int_{0}^{3} \sqrt{y+1} d y\) b. \(\int_{-1}^{0} \sqrt{y+1} d y\)

Evaluate the indefinite integrals by using the given substitutions to reduce the integrals to standard form. $$\int 2(2 x+4)^{5} d x, \quad u=2 x+4$$

Interpreting Limits of Sums as Integrals Express the limits in Exercises \(1-8\) as definite integrals. \(\lim _{|P| \rightarrow 0} \sum_{k=1}^{n} c_{k}^{2} \Delta x_{k},\) where \(P\) is a partition of [0,2]

Evaluate the integrals. $$\int_{0}^{2} x(x-3) d x$$

Without sigma notation. Then evaluate them. $$\sum_{k=1}^{2} \frac{6 k}{k+1}$$

Use finite approximations to estimate the area under the graph of the function using a. a lower sum with two rectangles of equal width. b. a lower sum with four rectangles of equal width. c. an upper sum with two rectangles of equal width. d. an upper sum with four rectangles of equal width. \(f(x)=x^{2}\) between \(x=0\) and \(x=1.\)

Evaluate the indefinite integrals by using the given substitutions to reduce the integrals to standard form. $$\int 7 \sqrt{7 x-1} d x, \quad u=7 x-1$$

Interpreting Limits of Sums as Integrals Express the limits in Exercises \(1-8\) as definite integrals. \(\lim _{|P| \rightarrow 0} \sum_{k=1}^{n} 2 c_{k}^{3} \Delta x_{k},\) where \(P\) is a partition of [-1,0]

Evaluate the integrals. $$\int_{-1}^{1}\left(x^{2}-2 x+3\right) d x$$

Without sigma notation. Then evaluate them. $$\sum_{k=1}^{3} \frac{k-1}{k}$$

Use finite approximations to estimate the area under the graph of the function using a. a lower sum with two rectangles of equal width. b. a lower sum with four rectangles of equal width. c. an upper sum with two rectangles of equal width. d. an upper sum with four rectangles of equal width. \(f(x)=x^{3}\) between \(x=0\) and \(x=1.\)

Use the Substitution Formula in Theorem 7 to evaluate the integrals in Exercises \(1-46\). a. \(\int_{0}^{\pi / 4} \tan x \sec ^{2} x d x\) b. \(\int_{-\pi / 4}^{0} \tan x \sec ^{2} x d x\)

Evaluate the indefinite integrals by using the given substitutions to reduce the integrals to standard form. $$\int 2 x\left(x^{2}+5\right)^{-4} d x, \quad u=x^{2}+5$$

Evaluate the integrals. $$\int_{-2}^{2} \frac{3}{(x+3)^{4}} d x$$

Without sigma notation. Then evaluate them. $$\sum_{k=1}^{4} \cos k \pi$$

Use finite approximations to estimate the area under the graph of the function using a. a lower sum with two rectangles of equal width. b. a lower sum with four rectangles of equal width. c. an upper sum with two rectangles of equal width. d. an upper sum with four rectangles of equal width. \(f(x)=1 / x\) between \(x=1\) and \(x=5.\)

Use the Substitution Formula in Theorem 7 to evaluate the integrals in Exercises \(1-46\). a. \(\int_{0}^{\pi} 3 \cos ^{2} x \sin x d x\) b. \(\int_{2 \pi}^{3 \pi} 3 \cos ^{2} x \sin x d x\)

Evaluate the indefinite integrals by using the given substitutions to reduce the integrals to standard form. $$\int \frac{4 x^{3}}{\left(x^{4}+1\right)^{2}} d x, u=x^{4}+1$$

Interpreting Limits of Sums as Integrals Express the limits in Exercises \(1-8\) as definite integrals. \(\lim _{|P| \rightarrow 0} \sum_{i=1}^{n}\left(\frac{1}{c_{i}}\right) \Delta x_{i},\) where \(P\) is a partition of [1,4]

Evaluate the integrals. $$\int_{-1}^{1} x^{299} d x$$

Without sigma notation. Then evaluate them. $$\sum_{k=1}^{5} \sin k \pi$$

Use finite approximations to estimate the area under the graph of the function using a. a lower sum with two rectangles of equal width. b. a lower sum with four rectangles of equal width. c. an upper sum with two rectangles of equal width. d. an upper sum with four rectangles of equal width. \(f(x)=4-x^{2}\) between \(x=-2\) and \(x=2.\)

Evaluate the indefinite integrals by using the given substitutions to reduce the integrals to standard form. $$\int(3 x+2)\left(3 x^{2}+4 x\right)^{4} d x, \quad u=3 x^{2}+4 x$$

Interpreting Limits of Sums as Integrals Express the limits in Exercises \(1-8\) as definite integrals. \(\lim _{W \rightarrow 0} \sum_{k=1}^{n} \frac{1}{1-c_{k}} \Delta x_{k},\) where \(P\) is a partition of [2,3]

Evaluate the integrals. $$\int_{1}^{4}\left(3 x^{2}-\frac{x^{3}}{4}\right) d x$$

Without sigma notation. Then evaluate them. $$\sum_{k=1}^{3}(-1)^{k+1} \sin \frac{\pi}{k}$$

Using rectangles each of whose height is given by the value of the function at the midpoint of the rectangle's base (the midpoint rule), estimate the area under the graphs of the following functions, using first two and then four rectangles. \(f(x)=x^{2}\) between \(x=0\) and \(x=1.\)

Use the Substitution Formula in Theorem 7 to evaluate the integrals in Exercises \(1-46\). a. \(\int_{0}^{\sqrt{7}} t\left(t^{2}+1\right)^{1 / 3} d t\) b. \(\int_{-\sqrt{7}}^{0} t\left(t^{2}+1\right)^{1 / 3} d t\)

Evaluate the indefinite integrals by using the given substitutions to reduce the integrals to standard form. $$\int \frac{(1+\sqrt{x})^{1 / 3}}{\sqrt{x}} d x, \quad u=1+\sqrt{x}$$

Interpreting Limits of Sums as Integrals Express the limits in Exercises \(1-8\) as definite integrals. \(\lim _{|R| \rightarrow 0} \sum_{k=1}^{n} \sqrt{4-c_{k}^{2}} \Delta x_{k},\) where \(P\) is a partition of [0,1]

Evaluate the integrals. $$\int_{-2}^{3}\left(x^{3}-2 x+3\right) d x$$

Without sigma notation. Then evaluate them. $$\sum_{k=1}^{4}(-1)^{k} \cos k \pi$$

Using rectangles each of whose height is given by the value of the function at the midpoint of the rectangle's base (the midpoint rule), estimate the area under the graphs of the following functions, using first two and then four rectangles. \(f(x)=x^{3}\) between \(x=0\) and \(x=1.\)

Evaluate the indefinite integrals by using the given substitutions to reduce the integrals to standard form. $$\int \sin 3 x \, d x, \quad u=3 x$$

Interpreting Limits of Sums as Integrals Express the limits in Exercises \(1-8\) as definite integrals. \(\lim _{[n] \rightarrow 0} \sum_{k=1}^{n}\left(\sec c_{k}\right) \Delta x_{k},\) where \(P\) is a partition of \([-\pi / 4,0]\)

Evaluate the integrals. $$\int_{0}^{1}\left(x^{2}+\sqrt{x}\right) d x$$

Which of the following express \(1+2+4+8+16+32\) in sigma notation? $$\begin{array}{lll} \text { a. } \sum_{k=1}^{6} 2^{k-1} & \text { b. } \sum_{k=0}^{5} 2^{k} & \text { c. } \sum_{k=-1}^{4} 2^{k+1} \end{array}$$

Using rectangles each of whose height is given by the value of the function at the midpoint of the rectangle's base (the midpoint rule), estimate the area under the graphs of the following functions, using first two and then four rectangles. \(f(x)=1 / x\) between \(x=1\) and \(x=5.\)

Use the Substitution Formula in Theorem 7 to evaluate the integrals in Exercises \(1-46\). a. \(\int_{0}^{1} \frac{10 \sqrt{v}}{\left(1+v^{3 / 2}\right)^{2}} d v\) b. \(\int_{1}^{4} \frac{10 \sqrt{v}}{\left(1+v^{3 / 2}\right)^{2}} d v\)

Evaluate the indefinite integrals by using the given substitutions to reduce the integrals to standard form. $$\int x \sin \left(2 x^{2}\right) d x, \quad u=2 x^{2}$$

Interpreting Limits of Sums as Integrals Express the limits in Exercises \(1-8\) as definite integrals. \(\lim _{|V| \rightarrow 0} \sum_{l=1}^{n}\left(\tan c_{l}\right) \Delta x_{k},\) where \(P\) is a partition of \([0, \pi / 4]\)

Evaluate the integrals. $$\int_{1}^{32} x^{-6 / 5} d x$$

Using rectangles each of whose height is given by the value of the function at the midpoint of the rectangle's base (the midpoint rule), estimate the area under the graphs of the following functions, using first two and then four rectangles. \(f(x)=4-x^{2}\) between \(x=-2\) and \(x=2.\)

Which of the following express \(1-2+4-8+16-32\) in sigma notation? a. \(\sum_{k=1}^{6}(-2)^{k-1}\) b. \(\sum_{k=0}^{5}(-1)^{k} 2^{k}\) c. \(\sum_{k=-2}^{3}(-1)^{k+1} 2^{k+2}\)

Suppose that \(f\) and \(g\) are integrable and that $$\int_{1}^{2} f(x) d x=-4, \quad \int_{1}^{5} f(x) d x=6, \quad \int_{1}^{5} g(x) d x=8$$ Use the rules in Table 5.6 to find a. \(\int_{2}^{2} g(x) d x\) b. \(\int_{5}^{1} g(x) d x\) c. \(\int_{1}^{2} 3 f(x) d x\) d. \(\int_{2}^{5} f(x) d x\) e. \(\int_{1}^{5}[f(x)-g(x)] d x\) f. \(\int_{1}^{5}[4 f(x)-g(x)] d x\)

Evaluate the indefinite integrals by using the given substitutions to reduce the integrals to standard form. $$\int \sec 2 t \tan 2 t \, d t, \quad u=2 t$$

Evaluate the integrals. $$\int_{0}^{\pi / 3} 2 \sec ^{2} x d x$$

Which formula is not equivalent to the other two? a. \(\sum_{k=2}^{4} \frac{(-1)^{k-1}}{k-1}\) b. \(\sum_{k=0}^{2} \frac{(-1)^{k}}{k+1}\) c. \(\sum_{k=-1}^{1} \frac{(-1)^{k}}{k+2}\)

Suppose that \(f\) and \(h\) are integrable and that $$\int_{1}^{9} f(x) d x=-1, \quad \int_{7}^{9} f(x) d x=5, \quad \int_{7}^{9} h(x) d x=4$$ a. \(\int_{1}^{9}-2 f(x) d x\) b. \(\int_{7}^{9}[f(x)+h(x)] d x\) c. \(\int_{7}^{9}[2 f(x)-3 h(x)] d x \quad\) d. \(\int_{9}^{1} f(x) d x\) e. \(\int_{1}^{7} f(x) d x \$ f. \)\int_{9}^{7}[h(x)-f(x)] d x$

Evaluate the indefinite integrals by using the given substitutions to reduce the integrals to standard form. $$\int\left(1-\cos \frac{t}{2}\right)^{2} \sin \frac{t}{2} d t, \quad u=1-\cos \frac{t}{2}$$