Problem 66
Completing the square Evaluate the following integrals. $$\int_{1}^{4} \frac{d t}{t^{2}-2 t+10}$$
Problem 66
Integrals of cot \(x\) and \(\csc x\) Use a change of variables to prove that \(\int \cot x \, d x=\ln |\sin x|+C\) (Hint: See Example \(1 .\) )
Problem 66
Use the reduction formulas in a table of integrals to maluate the following integrals. $$\int p^{2} e^{-3 p} d p$$
Problem 66
Integrating derivatives Use integration by parts to show that if \(f^{\prime}\) is continuous on \([a, b],\) then $$\int_{a}^{b} f(x) f^{\prime}(x) d x=\frac{1}{2}\left(f(b)^{2}-f(a)^{2}\right)$$.
Problem 67
Estimating error Refer to Theorem 8.1 in the following exercises. Let \(f(x)=\sqrt{x^{3}+1}\) a. Find a Midpoint Rule approximation to \(\int_{1}^{6} \sqrt{x^{3}+1} d x\) using \(n=50\) subintervals. b. Calculate \(f^{\prime \prime}(x)\) c. Use the fact that \(f^{\text {- }}\) is decreasing and positive on [1,6] to show that \(\left|f^{*}(x)\right| \leq 15 /(8 \sqrt{2})\) on [1,6] d. Use Theorem 8.1 to find an upper bound on the absolute error in the estimate found in part (a).
Problem 67
Use the reduction formulas in a table of integrals to maluate the following integrals. $$\int \tan ^{4} 3 y d y$$
Problem 67
Function defined as an integral Find the are length of the function \(f(x)=\int_{e}^{x} \sqrt{\ln ^{2} t-1} d t\) on \(\left[e, e^{3}\right]\).
Problem 67
Completing the square Evaluate the following integrals. $$\int \frac{x^{2}-8 x+16}{\left(9+8 x-x^{2}\right)^{3 / 2}} d x$$
Problem 67
Evaluate the following integrals. $$\int \frac{x^{2}}{\sqrt{1-9 x^{2}}} d x$$
Problem 68
Use the reduction formulas in a table of integrals to maluate the following integrals. $$\int \sec ^{4} 4 t d t$$
