Chapter 8

Round off your calculations to four decimal places. Estimate $$\int_{-1}^{12} x^{2} d x$$ by: (a) the len-cndpoint cstimate, \(n=12 ;\) (b) the right- endpoint estimate, \(n=12 ;(\mathrm{c})\) the midpoint estimate, \(n=6\) (d) the trapezoidal rule, \(n=12 ;\) (e) Simpson's rule, \(n=6\) Chcck your results by performing the integration.

Calculate. $$\int \frac{d x}{1-\sqrt{x}}$$

Decompose into partial fractions.$$\frac{1}{x^{2}+7 x+6}$$.

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

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

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

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

Calculate. $$\int_{5 / 2}^{4} \frac{x}{\sqrt{x^{2}-4}} d x$$.

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

Decompose into partial fractions.$$\frac{x^{2}}{(x-1)\left(x^{2}+4 x+5\right)}$$.

Calculate. $$\int_{0}^{2} x 2^{2} d x$$

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

Round off your calculations to four decimal places. Estimate $$\int_{0}^{3} \frac{d x}{1+x^{3}}$$ by: (a) the left-endpoint estimate, \(n=6 ;\) (b) the rightendpoint estimate, \(n=6 ;(\mathrm{c})\) the midpoint estimate, \(n=3\) (d) the trapezoidal rulc, \(n=6 ;\) (e) Simpson's rule, \(n=3\)

Calculate. $$\int \sqrt{x^{2}-1} d x$$.

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

Decompose into partial fractions.$$\frac{x}{x^{4}-1}$$.

Calculate. (If you run out of ideas, use the examples as models.) $$\int_{0}^{\pi / 4} \sin ^{2} 3 x d x$$.

Calculate. $$\int_{1}^{1} \sin \pi x d x$$

Round off your calculations to four decimal places. Estimate $$\int_{0}^{\pi} \frac{\sin x}{\pi+x} d x$$ by: (a) the trapezoidal rule, \(n=6 ;\) (b) Simpson's nule. \(n\) (Note the superiority of Simpson's rule.)

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

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

Decompose into partial fractions$$\frac{x^{4}}{(x-1)^{3}}$$.

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

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

Calculate. $$\int_{0}^{1} \operatorname{sec} \pi x \tan \pi x d x$$

Round off your calculations to four decimal places. Estimate the value of \(\pi\) by cotimating the integral $$\int_{0}^{1} \frac{d x}{1+x^{2}}=\arctan 1=\frac{\pi}{4}$$ by: (a) the trapezoidal rule, \(n=4 ;\) (b) Simpson's rule, \(n=2\)

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

Calculate. $$\int x \sqrt{1+x} d x$$

Decompose into partial fractions$$\frac{x^{2}-3 x-1}{x^{3}+x^{2}-2 x}$$.

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

Calculate. $$\int_{0}^{1} x^{2} e^{-x} d x$$

Calculate. $$\int \sec ^{2}(1-x) d x$$

Round off your calculations to four decimal places. Estimate $$\int_{0}^{2} \frac{d x}{\sqrt{4+x^{2}}}$$ by: (a) the trapezoidal rule, \(n=4 ;\) (b) Simpson's rule, \(n=2\)

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

Decompose into partial fractions$$\frac{x^{3}+x^{2}+x+2}{x^{4}+3 x^{2}+2}$$.

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

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

Calculate. $$\int \frac{d x}{5^{2}}$$

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

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

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

Decompose into partial fractions.$$\frac{2 x^{2}+1}{x^{3}-6 x^{2}+11 x-6}$$.

Calculate. $$\int(x+2) \sqrt{x-1} d x$$

Calculate. $$\int \frac{x^{2}}{\sqrt{1-x}} d x$$

Calculate. $$\int_{a / 6}^{\pi / 3} \cot x d x$$

Round off your calculations to four decimal places. Estimate $$\int_{-1}^{1} \cos x^{2} d x$$ by: (a) the midpoint estimate, \(n=4 ;\) (b) the trapezoidal rulc. \(n=8 .(\mathrm{c})\) Simpson's rule, \(n=4\)

Round off your calculations to four decimal places. Estimate $$\int_{1}^{2} \frac{e^{x}}{x} d x$$ by: (a) the midpoint estimate, \(n=4 ;\) (b) the trapezoidal rulc. \(n=\mathrm{S} .\) (c) Simpson's rule, \(n=4\)

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

Decompose into partial fractions.$$\frac{1}{x\left(x^{2}+1\right)^{2}}$$.

Calculate. (If you run out of ideas, use the examples as models.) $$\int \sin ^{2} x \cos ^{4} x d x$$.