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Chapter 2: Problem 3
Find the derivative of the function \(\mathrm{y}=\int_{0}^{\mathrm{x}} \frac{1-\mathrm{t}+\mathrm{t}^{2}}{1+\mathrm{t}+\mathrm{t}^{2}} \mathrm{dt}\) at \(\mathrm{x}=1\).
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If oil leaks from a tank at a rate of \(\mathrm{r}(\mathrm{t})\) litres per minute at time \(t\), what does \(\int_{0}^{120} r(t) d t\) represent?
Evaluate \(\int_{0}^{\pi / 2} \ln (1+\cos \theta \cos x) \frac{d x}{\cos x}\)
Find \(\int_{0}^{2} f(x) d x\), where
\(f(x)=\left\\{\begin{array}{l}\frac{1}{\sqrt[4]{x^{3}}} \quad \text { for } 0
\leq x \leq 1 \\ \frac{1}{\sqrt[4]{(x-1)^{3}}} & \text { for }
1
Sketch the region whose area is \(\int_{0}^{\infty} \frac{\mathrm{dx}}{1+\mathrm{x}^{2}}\), and use your sketch to show that \(\int_{0}^{\infty} \frac{\mathrm{dx}}{1+\mathrm{x}^{2}}=\int_{0}^{1} \sqrt{\frac{1-\mathrm{y}}{\mathrm{y}}} \mathrm{dy}\)
Prove that (i) \(\int_{1}^{\infty} \frac{\mathrm{dx}}{\left(\mathrm{x}+\sqrt{\mathrm{x}^{2}+1}\right)^{\mathrm{n}}}=\frac{\mathrm{n}}{\mathrm{n}^{2}-1}, \mathrm{n}>1\) (ii) \(\int_{1}^{\infty} \frac{d x}{\left(1+e^{x}\right)\left(1+e^{-x}\right)}=1\) (iii) \(\int_{0}^{\infty} \frac{x \ln x}{\left(1+x^{2}\right)^{2}} d x=0\) (iv) \(\int_{0}^{\infty} \frac{\sqrt{x}}{(1+x)^{2}} d x=\frac{1}{2}+\frac{1}{4} \pi\).
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