91Ó°ÊÓ

Are the statements true or false? Give an explanation for your answer. If \(F(x)=\int_{0}^{x} f(t) d t\) and \(G(x)=\int_{0}^{x} g(t) d t,\) then \(F(x)+G(x)=\int_{0}^{x}(f(t)+g(t)) d t\).

Evaluate the definite integrals exactly \([as in \)\ln (3 \pi)]\(,\) using the Fundamental Theorem, and numerically \([\ln (3 \pi) \approx 2.243]\) $$\int_{0}^{\pi / 4} \sin x d x$$

Are the statements true or false? Give an explanation for your answer. There is only one solution \(y(t)\) to the initial value problem \(d y / d t=3 t^{2}, y(1)=\pi\).

Evaluate the definite integrals exactly \([as in \)\ln (3 \pi)]\(,\) using the Fundamental Theorem, and numerically \([\ln (3 \pi) \approx 2.243]\) $$\int_{0}^{1} 2 e^{x} d x$$

Evaluate the definite integrals exactly \([as in \)\ln (3 \pi)]\(,\) using the Fundamental Theorem, and numerically \([\ln (3 \pi) \approx 2.243]\) $$\int_{0}^{2} 3 e^{x} d x$$

Evaluate the definite integrals exactly \([as in \)\ln (3 \pi)]\(,\) using the Fundamental Theorem, and numerically \([\ln (3 \pi) \approx 2.243]\) $$\int_{2}^{5}\left(x^{3}-\pi x^{2}\right) d x$$

Evaluate the definite integrals exactly \([as in \)\ln (3 \pi)]\(,\) using the Fundamental Theorem, and numerically \([\ln (3 \pi) \approx 2.243]\) $$\int_{0}^{1} \sin \theta d \theta$$

Evaluate the definite integrals exactly \([as in \)\ln (3 \pi)]\(,\) using the Fundamental Theorem, and numerically \([\ln (3 \pi) \approx 2.243]\) $$\int_{1}^{2} \frac{1+y^{2}}{y} d y$$

Evaluate the definite integrals exactly \([as in \)\ln (3 \pi)]\(,\) using the Fundamental Theorem, and numerically \([\ln (3 \pi) \approx 2.243]\) $$\int_{0}^{2}\left(\frac{x^{3}}{3}+2 x\right) d x$$

Evaluate the definite integrals exactly \([as in \)\ln (3 \pi)]\(,\) using the Fundamental Theorem, and numerically \([\ln (3 \pi) \approx 2.243]\) $$\int_{0}^{\pi / 4}(\sin t+\cos t) d t$$