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Magazine Problem 2
Evaluate the indefinite integrals by using the given substitutions to reduce the integrals to standard form. $$\int 7 \sqrt{7 x-1} d x, \quad u=7 x-1$$
Problem 3
Evaluate the indefinite integrals by using the given substitutions to reduce the integrals to standard form. $$\int 2 x\left(x^{2}+5\right)^{-4} d x, \quad u=x^{2}+5$$
Problem 3
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 \(x=5\)
Problem 3
Express the limits as definite integrals. $$\lim _{|P| \rightarrow 0} \sum_{k=1}^{n}\left(c_{k}^{2}-3 c_{k}\right) \Delta x_{k}, \text { where } P \text { is a partition of }[-7,5]$$
Problem 3
Evaluate the integrals. $$\int_{-2}^{2} \frac{3}{(x+3)^{4}} d x.$$
Problem 4
Evaluate the indefinite integrals by using the given substitutions to reduce the integrals to standard form. $$\int \frac{4 x^{3}}{\left(x^{4}+1\right)^{2}} d x, \quad u=x^{4}+1$$
Problem 4
Evaluate the integrals. $$\int_{-1}^{1} x^{299} d x.$$
Problem 4
Express the limits as definite integrals. $$\lim _{\|P\| \rightarrow 0} \sum_{k=1}^{n}\left(\frac{1}{c_{k}}\right) \Delta x_{k}, \text { where } P \text { is a partition of }[1,4]$$
Problem 4
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 \(x=2\)
Problem 5
Express the limits as definite integrals. $$\lim _{\|P\| \rightarrow 0} \sum_{k=1}^{n} \frac{1}{1-c_{k}} \Delta x_{k}, \text { where } P \text { is a partition of }[2,3]$$
