91Ó°ÊÓ

you evaluated \(\int \sin ^{2} \theta d \theta\) using integration by parts. (If you did not do it by parts, do so now!) Redo this integral using the identity \(\sin ^{2} \theta=\) \((1-\cos 2 \theta) / 2 .\) Explain any differences in the form of the answer obtained by the two methods.

Evaluate the indefinite integral, using a trigonometric substitution and a triangle to express the answer in terms of \(x\).$$\int \frac{1}{x \sqrt{9-4 x^{2}}} d x$$

Evaluate the definite integrals. Whenever possible, use the Fundamental Theorem of Calculus, perhaps after a substitution. Otherwise, use numerical methods. $$\int_{0}^{1} \sqrt{3-x^{2}} d x$$

Find the general antiderivative. Check your answers by differentiation. $$f(x)=\frac{1}{3 \cos ^{2}(2 x)}$$

Evaluate the indefinite integral, using a trigonometric substitution and a triangle to express the answer in terms of \(x\).$$\int \frac{1}{x \sqrt{1+16 x^{2}}} d x$$.

Compute \(\int \cos ^{2} \theta d \theta\) in two different ways and explain any differences in the form of your answers. (The identity \(\left.\cos ^{2} \theta=(1+\cos 2 \theta) / 2 \text { may be useful. }\right)\)

Evaluate the definite integrals. Whenever possible, use the Fundamental Theorem of Calculus, perhaps after a substitution. Otherwise, use numerical methods. $$\int_{0}^{1} \frac{1}{x^{2}+2 x+1} d x$$

Evaluate the indefinite integral, using a trigonometric substitution and a triangle to express the answer in terms of \(x\).$$\int \frac{1}{x^{2} \sqrt{4-x^{2}}} d x$$.

Decide whether the statements are true or false. Give an explanation for your answer. If \(f\) is continuous for all \(x\) and \(\int_{0}^{\infty} f(x) d x\) converges, then so does \(\int_{a}^{\infty} f(x) d x\) for all positive \(a\)

Use the Fundamental Theorem to calculate the definite integrals. $$\int_{0}^{1 / 2} \cos (\pi x) d x$$