91Ó°ÊÓ

Decide whether the statements are true or false. Give an explanation for your answer. If \(\int_{0}^{\infty} f(x) d x\) and \(\int_{0}^{\infty} g(x) d x\) both converge, then \(\int_{0}^{\infty}(f(x)+g(x)) d x\) converges.

Using properties of ln, find a substitution \(w\) and constants \(k, n\) so that the integral has the form. $$\int k w^{n} \ln w d w$$ $$\int(2 x+1)^{3} \ln (2 x+1) d x$$

Evaluate the indefinite integral, using a trigonometric substitution and a triangle to express the answer in terms of \(x\).$$\int \frac{x^{3}}{\sqrt{4-x^{2}}} d x$$.

Decide whether the statements are true or false. Give an explanation for your answer. If \(\int_{0}^{\infty} f(x) d x\) and \(\int_{0}^{\infty} g(x) d x\) both diverge, then \(\int_{0}^{\infty}(f(x)+g(x)) d x\) diverges.

Use the Fundamental Theorem to calculate the definite integrals. $$\int_{1}^{4} \frac{\cos \sqrt{x}}{\sqrt{x}} d x$$

If \(f\) is a twice differentiable function, find \(\int x f^{\prime \prime}(x) d x\) (Your answer should contain \(f,\) but no integrals.)

Suppose that \(f\) is continuous for all real numbers and that \(\int_{0}^{\infty} f(x) d x\) converges. Let \(a\) be any positive number. Decide which of the statements are true and which are false. Give an explanation for your answer. $$\int_{0}^{\infty} a f(x) d x \text { converges. }$$

Using properties of ln, find a substitution \(w\) and constants \(k, n\) so that the integral has the form. $$\int k w^{n} \ln w d w$$ $$\int(2 x+1)^{3} \ln \frac{1}{\sqrt{2 x+1}} d x$$

Decide whether the statements are true for all continuous functions, \(f\). Give an explanation for your answer. If \(\operatorname{LEFT}(2)<\int_{a}^{b} f(x) d x,\) then \(\operatorname{LEFT}(4)<\) \(\int_{a}^{b} f(x) d x.\)

Find the exact area of the regions.Bounded by \(y=3 x /((x-1)(x-4)), y=0, x=2\) \(x=3\).