Chapter 8

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

Calculate. $$\int \frac{d x}{x(\ln x)^{3}}$$

Calculate. $$\int_{0}^{1} \frac{x^{3}}{1+x^{4}} d x$$

Round off your calculations to four decimal places. Estimate $$\int_{0}^{2} e^{-x^{2}} d x$$ by: (a) the tranezoidal rule, \(n=10 ;\) (b) Simpson's rulc. \(n=5\)

Calculate. $$\int_{0}^{1 / 2} \frac{x^{2}}{\left(1-x^{2}\right)^{3 / 2}} d x$$.

Calculate.$$\int \frac{7}{(x-2)(x+5)} d x$$.

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

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

Calculate. $$\int_{1}^{e^{2}} x \ln \sqrt{x} d x$$

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

Round off your calculations to four decimal places. Estimate $$\int_{1}^{4} \frac{1}{\ln x} d x$$ by: (a) the midpoint estimate, \(n=4 ;\) (b) the trapczoidal rule, \(n=8 ;(\mathrm{c})\) Simpson's re!e, \(n=4\)

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

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

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

Calculate. $$\int x(1+x)^{1 / 3} d x$$

Calculate. $$\int_{0}^{3} x \sqrt{x+1} d x$$

Calculate. $$\int_{-\pi / 4}^{\pi / 4} \frac{d x}{\cos ^{2} x}$$

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

Calculate.$$\int \frac{2 x^{4}-4 x^{3}+4 x^{2}+3}{x^{3}-x^{2}} d x$$.

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

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

Calculate. $$\int_{-\pi / 4}^{\pi / 4} \frac{\sin x}{\cos ^{2} x} d x$$

Show that there is a unique parabola of the form \(y=\) \(A x^{2}+B x+C\) through three distinct noncollincar points with different \(x\) -coordinates.

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

Show that the function \(g(x)=A x^{2}+B x+C\) satisfies the condition $$\int_{a}^{b} g(x) d x=\frac{b-a}{6}\left[g(a)+4 g\left(\frac{a+b}{2}\right)+g(b)\right]$$ for every interval \([a, b]\)

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

Calculate. $$\int_{0}^{2} \frac{x^{2}}{\sqrt{16-x^{2}}} d x$$.

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

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

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

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

Determine the values of \(n\) which guarante: a theoretical error less than \(\epsilon\) if the integral is estimated by: \((a)\) the trapezoidal rule; (b) Simpson's rule. $$\int_{1}^{4} \sqrt{x} d x ; \quad \epsilon=0.01$$

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

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

Calculate. $$\int_{0}^{5} x^{2} \sqrt{25-x^{2}} d x$$.

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

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

Calculate. $$\int_{1}^{2} \frac{e^{1 / x}}{x^{2}} d x$$

Determine the values of \(n\) which guarante: a theoretical error less than \(\epsilon\) if the integral is estimated by: \((a)\) the trapezoidal rule; (b) Simpson's rule. $$\int_{1}^{3} x^{2} d x ; \quad \epsilon=0.01$$

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

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

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

Calculate. $$\int x(x+5)^{-14} d x$$

Calculate. $$\int \frac{x^{3}}{\sqrt{1-x^{4}}} d x$$

Determine the values of \(n\) which guarante: a theoretical error less than \(\epsilon\) if the integral is estimated by: \((a)\) the trapezoidal rule; (b) Simpson's rule. $$\int_{1}^{4} \sqrt{x} d x ; \quad \epsilon=0.00001$$

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

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

Calculate. $$\int \frac{x^{2}}{\left(x^{2}+8\right)^{3 / 2}} d x$$.

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

Calculate. $$\int_{0}^{e} \frac{d x}{x^{2}+c^{2}}$$