TEST 14
1.
\(\displaystyle\int f'(5x+7)\,dx=?\)
A) \(f(5x+7)+c\)B) \(5\cdot f(5x+7)+c\)C) \(f(5)+c\)D) \(\samefrac{x^2}{2}+\ln|x|+\samefrac1x+c\)E) \(\samefrac{1}{5}\cdot f(5x+7)+c\)
2.
\(A=\displaystyle\int \samefrac{e^{2x}+1}{e^x-1}\,dx\)
\(u=e^x\)
\(\Rightarrow A=?\)
A) \(\displaystyle\int \samefrac{u^2+1}{u-1}\,du\)B) \(\displaystyle\int \samefrac{u^3+u}{u-1}\,du\)C) \(\displaystyle\int \samefrac{u^3+u}{u^2-1}\,du\)D) \(\displaystyle\int \samefrac{u^2+1}{u^2-1}\,du\)E) \(\displaystyle\int \samefrac{u^2+1}{u^2-u}\,du\)
3.
\(A=\displaystyle\int \samefrac{\sqrt[3]{x+1}+\sqrt{x+1}}{x+1}\,dx\)
\(u=\sqrt[6]{x+1}\)
\(\Rightarrow A=?\)
A) \(\displaystyle\int \left(\samefrac1{u^4}+\samefrac1{u^3}\right)\,du\)B) \(\displaystyle\int (u^2+u^3)\,du\)C) \(6\displaystyle\int (u+u^2)\,du\)D) \(\displaystyle\int (u+u^2)\,du\)E) \(\displaystyle\int \left(\samefrac1{u+u^2}\right)\,du\)
4.
\(A=\displaystyle\int \samefrac1{9+x^2}\,dx\)
\(x=3\cdot\tan t\)
\(\Rightarrow A=?\)
A) \(\samefrac{1}{3}\displaystyle\int dt\)B) \(\displaystyle\int dt\)C) \(3\displaystyle\int dt\)D) \(9\displaystyle\int dt\)E) \(\samefrac{1}{9}\displaystyle\int dt\)
5.
\(A=\displaystyle\int \samefrac1{\cos^4x}\,dx\)
\(u=\tan x\)
\(\Rightarrow A=?\)
A) \(\displaystyle\int (u^2+1)\,du\)B) \(\displaystyle\int \samefrac1{u^2+1}\,du\)C) \(\displaystyle\int (u^2-1)\,du\)D) \(\displaystyle\int \samefrac1{u^4+u}\,du\)E) \(\displaystyle\int (u^4+1)\,du\)
6.
\(0\leq t<\samefrac{\pi}{2}\)
\(A=\displaystyle\int \sqrt{4-x^2}\,dx\)
\(x=2\sin t\)
\(\Rightarrow A=?\)
A) \(4\displaystyle\int (\cos^2t)\,dt\)B) \(-4\displaystyle\int (\cos^2t)\,dt\)C) \(4\displaystyle\int (\sin^2t)\,dt\)D) \(\displaystyle\int (\sin^2t)\,dt\)E) \(\displaystyle\int (\cos t)\,dt\)
7.
\(\displaystyle\int(x^3+x)d(x^2)=?\)
A) \(\samefrac{x^5}{5}+\samefrac{x^3}{3}+c\)B) \(\samefrac{x^6}{6}+\samefrac{x^4}{4}+c\)C) \(\samefrac{x^6}{3}+\samefrac{x^4}{2}+c\)D) \(\samefrac{2x^5}5+\samefrac{2x^3}3+c\)E) \(\samefrac{x^4}{4}+\samefrac{x^2}{2}+c\)
8.
\(0\leq x\leq\samefrac{\pi}{2}\)
\(\Rightarrow \displaystyle\int \sqrt{1-\cos x}\,dx=?\)
A) \(\sqrt2\cdot\cos\left(\samefrac x2\right)+c\)B) \(-\sqrt2\cdot\cos\left(\samefrac x2\right)+c\)C) \(-2\sqrt2\cdot\cos\left(\samefrac x2\right)+c\)D) \(-\samefrac{\sqrt2}2\cdot\cos x+c\)E) \(\samefrac{\sqrt2}2\cdot\cos x+c\)