Chapter 8

Calculate. $$\int \frac{d x}{\left(x^{2}+16\right)^{2}}$$

Calculate. (If you run out of ideas, use the examples as models.) $$\int \tan ^{6} x d x$$.

Calculate. $$\int \frac{d x}{e^{x} \sqrt{4+e^{2 x}}}$$.

Calculate. $$\int \frac{1}{2+\cos x} d x$$

Calculate. $$\int\left(e^{x}+2 x\right)^{2} d x$$

Calculate. $$\int \frac{e^{x}}{1+e^{2 x}} d x$$

Show that the trapezoidal rule is exact (thcorctical error zero) if / is lincar.

$$\text { (a) } \operatorname{Sc} f(x)=x^{2} \cdot \operatorname{Let}[a, b]=[0,1]$$ and take \(n=2 .\) Show that in this case the theorctical error inequality $$\left|E_{n}^{T}\right| \leq \frac{(b-a)^{3}}{12 n^{2}} M$$

Calculate. $$\int \frac{d x}{x^{4}+4}$$

Calculate. (If you run out of ideas, use the examples as models.) $$\int \cot ^{3} x \csc ^{3} x d x$$.

Calculate. $$\int \frac{d x}{x^{2} \sqrt{x^{2}-a^{2}}}$$.

Calculate. $$\int \frac{1}{2+\sin x} d x$$

Calculate. $$\int_{0}^{1} \ln \left(1+x^{2}\right) d x$$

Calculate. $$\int \frac{d x}{x^{2}+6 x+10}$$

Show that, if \(f\) is continuous. then \(T_{n}\) and \(S_{n}\) can both be Written as Ricmann sums.

Calculate. $$\int \frac{d x}{x^{4}+16}$$

Calculate. (If you run out of ideas, use the examples as models.) $$\int \tan ^{3} x \sec ^{3} x d x$$.

Calculate. $$\int \frac{e^{x}}{\sqrt{9-e^{2 x}}} d x$$.

Calculate. $$\int \frac{\sin x}{1+\sin ^{2} x} d x$$

Calculate. $$\int x \ln (x+1) d x$$

Calculate. $$\int e^{x} \tan e^{x} d x$$

Calculate. $$\int \frac{x-3}{x^{3}+x^{2}} d x$$

Calculate. (If you run out of ideas, use the examples as models.) $$\int \sin 5 x \sin 2 x d x$$.

Calculate. $$\int \frac{d x}{e^{x} \sqrt{e^{2 x}-9}}$$.

Calculate. $$\int \frac{1}{\sin x+\tan x} d x$$

Calculate. $$\int x \sin x^{2} d x$$

Calculate. $$\int \frac{1}{(x-1)\left(x^{2}+1\right)^{2}} d x$$

Calculate. (If you run out of ideas, use the examples as models.) $$\int \sec ^{4} 3 x d x$$.

Calculate. $$\int \frac{d x}{\sqrt{x^{2}-2 x-3}}$$.

Calculate. $$\int \frac{1}{1+\sin x+\cos x} d x$$

Calculate. $$\int e^{3 x} \cos 2 x d x$$

Calculate. $$\int \frac{x}{9+x^{4}} d x$$

Cire a \(C \wedge S\) and the trapezoidal rule to estimate: (a) \(\int_{0}^{16}(x+\cos x) d x, \quad n=50\) (b) \(\int_{-4}^{7}\left(x^{5}-5 x^{4}+x^{3}-3 x^{2}-x+4\right) d x, \quad n=30\)

Calculate. $$\int \frac{x+1}{x^{3}+x^{2}-6 x} d x$$

Calculate. (If you run out of ideas, use the examples as models.) $$\int \sin ^{5 / 2} x \cos ^{3} x d x$$.

Calculate. $$\int \frac{d x}{\left(x^{2}-4 x+4\right)^{3 / 2}}$$.

Calculate. $$\int \frac{1-\cos x}{1+\sin x} d x$$

Calculate. $$\int x^{3} \sin x^{2} d x$$

Calculate. $$\int \tan ^{2} x d x$$

Use a CAS and Simpson's :ule to estimate: (a) \(\int_{-4}^{3} \frac{x^{2}}{x^{2}+4} d x, \quad n=50\) (b) \(\int_{0}^{\pi / 6}(x+\tan x) d x, \quad n=25\)

Calculate. $$\int \frac{x^{3}+x^{2}+x+3}{\left(x^{2}+1\right)\left(x^{2}+3\right)} d x$$

Calculate. (If you run out of ideas, use the examples as models.) $$\int \frac{\sin ^{3} x}{\cos x} d x$$.

Calculate. $$\int \frac{x}{\sqrt{6 x-x^{2}}} d x$$.

Calculate. $$\int \frac{1}{5+3 \sin x} d x$$

Calculate. $$\int x^{3} \sin x d x$$

Calculate. $$\int \cosh 2 x \sinh ^{3} 2 x d x$$

Estimate the theoretical crror if Simpson's rule with \(n=20\) is used to approximate $$\int_{1}^{5} \frac{x^{2}-4}{x^{2}+9} d x$$

Evaluate. $$\int_{0}^{2} \frac{x}{x^{2}+5 x+6} d x$$

Calculate. (If you run out of ideas, use the examples as models.) $$\int \tan ^{5} 3 x d x$$.

Calculate. $$\int x \sqrt{6 x-x^{2}-8} d x$$.