Chapter 8

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

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. $$\int_{1}^{3} x^{5} d x ; \quad \epsilon=0.00001$$

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

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

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

Calculate. $$\int \sqrt{x} \ln x d x$$

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

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

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

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

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

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

Calculate. $$\int \frac{\sec ^{2} \theta}{\sqrt{3 \tan \theta+1}} d \theta$$

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

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

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

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

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

Calculate. $$\int \frac{\sin \phi}{3-2 \cos \phi} d \phi$$

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. $$\int_{1}^{3} e^{x} d x ; \quad \epsilon=0.01$$

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

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

Calculate. $$\int_{0}^{3} \frac{x^{3}}{\sqrt{9+x^{2}}} d x$$.

Calculate. $$\int 2 x^{2}(4 x+1)^{-12} d x$$

Calculate. $$\int_{0}^{1 / 2} x \cos \pi x d x$$

Calculate. $$\int \frac{e^{x}}{a e^{x}-b} d x$$

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

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

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

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

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

Calculate. $$\int \frac{d x}{x^{2}-4 x+13}$$

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

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

Calculate. $$\int \frac{x}{(a x+b)^{3 / 2}} d x$$

Calculate. $$\int x^{2}(x+1)^{9} d x$$

Calculate. $$\int \frac{x}{(x+1)^{2}+4} d x$$

Calculate. $$\int \frac{3 x^{5}-3 x^{2}+x}{x^{3}-1} d x$$

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

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

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

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

Calculate. $$\int \frac{\ln x}{x} d x$$

Show that Simpson's rule is exact (theorctical error zero) for every polynomial of degree 3 or less.

Calculate. $$\int \frac{x^{3}+4 x^{2}-4 x-1}{\left(x^{2}+1\right)^{2}} d x$$

Calculate. (If you run out of ideas, use the examples as models.) $$\int_{\pi / 6}^{\pi / 2} \cot ^{2} x d x$$.

Calculate. $$\int_{0}^{1} \frac{d x}{\left(5-x^{2}\right)^{3 / 2}}$$.

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

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

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