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Chapter 5: Problem 8
Symmetry in integrals Use symmetry to evaluate the following integrals. $$\int_{-200}^{200} 2 x^{5} d x$$
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Determine whether the following statements are true and give an explanation or counterexample. a. Suppose that \(f\) is a positive decreasing function, for \(x>0\) Then the area function \(A(x)=\int_{0}^{x} f(t) d t\) is an increasing function of \(x\) b. Suppose that \(f\) is a negative increasing function, for \(x>0\) Then the area function \(A(x)=\int_{0}^{x} f(t) d t\) is a decreasing function of \(x\) c. The functions \(p(x)=\sin 3 x\) and \(q(x)=4 \sin 3 x\) are antiderivatives of the same function. d. If \(A(x)=3 x^{2}-x-3\) is an area function for \(f,\) then \(B(x)=3 x^{2}-x\) is also an area function for \(f\) e. \(\frac{d}{d x} \int_{a}^{b} f(t) d t=0\)
Additional integrals Use a change of variables to evaluate the following integrals. $$\int_{0}^{\pi / 4} e^{\sin ^{2} x} \sin 2 x d x$$
General results Evaluate the following integrals in which the function \(f\) is unspecified. Note \(f^{(p)}\) is the pth derivative of \(f\) and \(f^{p}\) is the pth power of \(f\). Assume \(f\) and its derivatives are continuous for all real numbers. $$\int 2\left(f^{2}(x)+2 f(x)\right) f(x) f^{\prime}(x) d x$$
Find the area of the following regions. The region bounded by the graph of \(f(\theta)=\cos \theta \sin \theta\) and the \(\theta\) -axis between \(\theta=0\) and \(\theta=\pi / 2\).
Use the Fundamental Theorem of Calculus, Part \(1,\) to find the function \(f\) that satisfies the equation $$\int_{0}^{x} f(t) d t=2 \cos x+3 x-2$$ Verify the result by substitution into the equation.
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