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Magazine Problem 4
Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. $$\int_{\pi / 4}^{\pi / 3} \frac{d x}{\cos ^{2} x \tan x}$$
Problem 4
Evaluate the integrals. \(\int_{0}^{2} \frac{d x}{8+2 x^{2}}\)
Problem 4
The instructions for the integrals have two parts, one for the Trapezoidal Rule and one for Simpson's Rule. I. Using the Trapezoidal Rule a. Estimate the integral with \(n=4\) steps and find an upper bound for \(\left|E_{T}\right|\) b. Evaluate the integral directly and find \(\left|E_{T}\right|\) c. Use the formula \(\left(\left|E_{T}\right| /(\text { true value })\right) \times 100\) to express \(\left|E_{T}\right|\) as a percentage of the integral's true value. II. Using Simpson's Rule a. Estimate the integral with \(n=4\) steps and find an upper bound for \(\left|E_{S}\right|\) b. Evaluate the integral directly and find \(\left|E_{S}\right|\) c. Use the formula \(\left(\left|E_{S}\right| /(\text { true value })\right) \times 100\) to express \(\left|E_{S}\right|\) as a percentage of the integral's true value. $$\int_{-2}^{0}\left(x^{2}-1\right) d x$$
Problem 5
Evaluate the integrals using integration by parts. $$\int_{1}^{2} x \ln x d x$$
Problem 5
Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. $$\int \frac{1-x}{\sqrt{1-x^{2}}} d x$$
Problem 5
Use the table of integrals at the back of the book to evaluate the integrals. $$\int x \sqrt{2 x-3} d x$$
Problem 5
The instructions for the integrals have two parts, one for the Trapezoidal Rule and one for Simpson's Rule. I. Using the Trapezoidal Rule a. Estimate the integral with \(n=4\) steps and find an upper bound for \(\left|E_{T}\right|\) b. Evaluate the integral directly and find \(\left|E_{T}\right|\) c. Use the formula \(\left(\left|E_{T}\right| /(\text { true value })\right) \times 100\) to express \(\left|E_{T}\right|\) as a percentage of the integral's true value. II. Using Simpson's Rule a. Estimate the integral with \(n=4\) steps and find an upper bound for \(\left|E_{S}\right|\) b. Evaluate the integral directly and find \(\left|E_{S}\right|\) c. Use the formula \(\left(\left|E_{S}\right| /(\text { true value })\right) \times 100\) to express \(\left|E_{S}\right|\) as a percentage of the integral's true value. $$\int_{0}^{2}\left(t^{3}+t\right) d t$$
Problem 5
Evaluate the integrals. \(\int_{0}^{3 / 2} \frac{d x}{\sqrt{9-x^{2}}}\)
Problem 5
Evaluate the integrals. $$\int \sin ^{3} x d x$$
Problem 5
Expand the quotients by partial fractions. $$\frac{z+1}{z^{2}(z-1)}$$
