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TEST 24

1.
\(\displaystyle\int\limits_{1}^{e} \ln x\,dx=?\)
A) \(e\)B) \(1\)C) \(0\)D) \(-1\)E) \(-e\)
2.
\(\displaystyle\int\limits_{1}^{\sqrt3} \samefrac2{1+x^2}\,dx=?\)
A) \(\samefrac{\pi}{2}\)B) \(\samefrac{\pi}{3}\)C) \(\samefrac{\pi}{4}\)D) \(\samefrac{\pi}{6}\)E) \(\samefrac{\pi}{12}\)
3.
\(\displaystyle\int\limits_b^a dx=5\)
\(a+b=13\)
\(\Rightarrow a\cdot b=?\)
A) \(12\)B) \(18\)C) \(24\)D) \(30\)E) \(36\)
4.
\(\left.\begin{aligned}\displaystyle\displaystyle\int\limits_{1}^{5} f(x)\,dx=17\\\displaystyle\displaystyle\int\limits_{1}^{7} f(x)\,dx=30\end{aligned}\right\}\)
\(\Rightarrow \displaystyle\int\limits_{5}^{7} f(x)\,dx=?\)
A) \(-47\)B) \(-13\)C) \(13\)D) \(22\)E) \(47\)
5.
\(\samefrac d{dx}\left(\displaystyle\int\limits_{1}^{5} \sqrt{3x-5}\,dx\right)=?\)
A) \(-2\)B) \(0\)C) \(7\)D) \(10\)E) \(25\)
6.
\(f(x)=\samefrac{3x+2}{x-4}\)
\(\Rightarrow\displaystyle\int\limits_4^5d(f^{-1}(x))=?\)
A) \(30\)B) \(18\)C) \(-6\)D) \(-7\)E) \(-9\)
7.
\(A=\displaystyle\int\limits_{\samefrac{1}{2}}^{\samefrac{\sqrt3}2} \samefrac1{\sqrt{1-x^2}}\,dx\)
\(x=\sin u\)
\(\Rightarrow A=?\)
A) \(\displaystyle\int\limits_{\samefrac{\pi}{6}}^{\samefrac{\pi}{3}}du\)B) \(\displaystyle\int\limits_{\samefrac{\pi}{3}}^{\samefrac{\pi}{6}}du\)C) \(\displaystyle\int\limits_{\samefrac{1}{2}}^{\samefrac{\pi}{2}} u\,du\)D) \(\displaystyle\int\limits_{\samefrac{1}{2}}^{\samefrac{\sqrt3}2}du\)E) \(\displaystyle\int\limits_{\samefrac{\pi}{3}}^{\samefrac{\pi}{6}} \samefrac1{\cos u}\,du\)
8.
\(A=\displaystyle\int\limits_{1}^{9} \samefrac{\sqrt x+1}{2\sqrt x}\,dx\)
\(u=\sqrt x+1\)
\(\Rightarrow A=?\)
A) \(\displaystyle\int\limits_{2}^{4} u\,du\)B) \(\displaystyle\int\limits_1^9du\)C) \(\displaystyle\int\limits_{1}^{9} \samefrac u2\,du\)D) \(\displaystyle\int\limits_{2}^{4} \samefrac u2\,du\)E) \(\displaystyle\int\limits_{1}^{3} \samefrac{u^2}2\,du\)