Problem 68
Estimating error Refer to Theorem 8.1 in the following exercises. Let \(f(x)=e^{x^{2}}\) a. Find a Trapezoid Rule approximation to \(\int_{0}^{1} e^{x^{2}} d x\) using \(n=50\) subintervals. b. Calculate \(f^{-}(x)\) c. Explain why \(\left|f^{*}(x)\right|<18\) on [0,1] , given that \(e<3\). d. Use Theorem 8.1 to find an upper bound on the absolute error in the estimate found in part (a).
Problem 68
67-70. Integrals of the form \(\int \sin m x\) cos nux dx Use the following three identities to evaluate the given integrals. $$ \begin{aligned} \sin m x \sin n x &=\frac{1}{2}(\cos ((m-n) x)-\cos ((m+n) x)) \\ \sin m x \cos n x &=\frac{1}{2}(\sin ((m-n) x)+\sin ((m+n) x)) \\\ \cos m x \cos n x &=\frac{1}{2}(\cos ((m-n) x)+\cos ((m+n) x)) \end{aligned} $$ $$\int \sin 5 x \sin 7 x \, d x$$
Problem 68
Completing the square Evaluate the following integrals. $$\int \frac{d x}{\sqrt{(x-1)(3-x)}}$$
Problem 68
Log integrals Use integration by parts to show that for \(m \neq-1\) $$\int x^{m} \ln x d x=\frac{x^{m+1}}{m+1}\left(\ln x-\frac{1}{m+1}\right)+C$$ and for \(m=-1\) $$\int \frac{\ln x}{x} d x=\frac{1}{2} \ln ^{2} x+C$$.
Problem 68
Evaluate the following integrals. $$\int \frac{e^{x}}{e^{2 x}+2 e^{x}+17} d x$$
Problem 68
Different methods a. Evaluate \(\int \cot x \csc ^{2} x d x\) using the substitution \(u=\cot x\) b. Evaluate \(\int \cot x \csc ^{2} x d x\) using the substitution \(u=\csc x\) c. Reconcile the results in parts (a) and (b).
Problem 69
Estimating error Refer to Theorem 8.1 in the following exercises. Let \(f(x)=\sin e^{x}\) a. Find a Trapezoid Rule approximation to \(\int_{0}^{1} \sin e^{x} d x\) using \(n=40\) subintervals. b. Calculate \(f^{-\prime}(x)\) c. Explain why \(\left|f^{\prime \prime}(x)\right|<6\) on \([0,1],\) given that \(e<3\) (Hint: Graph \(f^{\star}\),) d. Find an upper bound on the absolute error in the estimate found in part (a) using Theorem 8.1.
Problem 69
Completing the square Evaluate the following integrals. $$\int_{2+\sqrt{2}}^{4} \frac{d x}{\sqrt{(x-1)(x-3)}}$$
Problem 69
67-70. Integrals of the form \(\int \sin m x\) cos nux dx Use the following three identities to evaluate the given integrals. $$ \begin{aligned} \sin m x \sin n x &=\frac{1}{2}(\cos ((m-n) x)-\cos ((m+n) x)) \\ \sin m x \cos n x &=\frac{1}{2}(\sin ((m-n) x)+\sin ((m+n) x)) \\\ \cos m x \cos n x &=\frac{1}{2}(\cos ((m-n) x)+\cos ((m+n) x)) \end{aligned} $$ $$\int \sin 3 x \sin 2 x \, d x$$
Problem 69
Use a computer algebra system to solve the following problems. Find the exact area of the region bounded by the curves \(y=\sqrt{x+\sqrt{x}}\) and \(y=\frac{2}{\sqrt{1+\sqrt{x}}}\) in the first quadrant.
