高數(shù)積分考試題及答案

單項(xiàng)選擇題（每題2分，共10題）1.下列函數(shù)中，哪個(gè)是\(f(x)=x^2\)的一個(gè)原函數(shù)？A.\(x^3\)B.\(\frac{1}{3}x^3\)C.\(3x^3\)D.\(\frac{3}{x^3}\)2.\(\intxdx=\)？A.\(\frac{1}{2}x^2\)B.\(x^2\)C.\(\frac{1}{2}x\)D.\(2x\)3.\(\int_{0}^{1}x^2dx=\)？A.\(\frac{1}{3}\)B.\(\frac{1}{2}\)C.\(1\)D.\(3\)4.若\(F^\prime(x)=f(x)\)，則\(\intf(2x)dx=\)？A.\(F(2x)+C\)B.\(\frac{1}{2}F(2x)+C\)C.\(2F(2x)+C\)D.\(F(x)+C\)5.\(\int\sinxdx=\)？A.\(\cosx\)B.\(-\cosx\)C.\(\sinx\)D.\(-\sinx\)6.\(\int_{1}^{e}\frac{1}{x}dx=\)？A.\(1\)B.\(e\)C.\(0\)D.\(-1\)7.已知\(f(x)\)的一個(gè)原函數(shù)是\(e^x\)，則\(f(x)=\)？A.\(e^x\)B.\(e^x+C\)C.\(xe^x\)D.\(e^x+x\)8.\(\int(2x+3)dx=\)？A.\(x^2+3x\)B.\(x^2+3x+C\)C.\(2x^2+3x\)D.\(2x^2+3x+C\)9.\(\int_{0}^{\frac{\pi}{2}}\cosxdx=\)？A.\(0\)B.\(1\)C.\(-1\)D.\(\frac{\pi}{2}\)10.下列積分中，值為\(0\)的是？A.\(\int_{-1}^{1}x^2dx\)B.\(\int_{-1}^{1}xdx\)C.\(\int_{0}^{1}xdx\)D.\(\int_{0}^{1}x^2dx\)答案：1.B2.A3.A4.B5.B6.A7.A8.B9.B10.B多項(xiàng)選擇題（每題2分，共10題）1.下列函數(shù)中，是\(f(x)=2x\)原函數(shù)的有？A.\(x^2\)B.\(x^2+1\)C.\(x^2+C\)D.\(2x^2\)2.下列積分計(jì)算正確的有？A.\(\int_{0}^{1}1dx=1\)B.\(\int_{0}^{1}xdx=\frac{1}{2}\)C.\(\int_{0}^{1}x^2dx=\frac{1}{3}\)D.\(\int_{-1}^{1}x^3dx=0\)3.若\(F(x)\)是\(f(x)\)的原函數(shù)，則？A.\(F^\prime(x)=f(x)\)B.\(\intf(x)dx=F(x)+C\)C.\(f(x)\)的不定積分是\(F(x)\)D.\(F(x)\)是\(f(x)\)的唯一原函數(shù)4.下列積分中，結(jié)果為\(e-1\)的是？A.\(\int_{0}^{1}e^xdx\)B.\(\int_{1}^{e}1dx\)C.\(\int_{0}^{1}(e^x+1)dx\)D.\(\int_{0}^{1}e^x-1dx\)5.已知\(f(x)\)的一個(gè)原函數(shù)是\(\lnx\)，則？A.\(f(x)=\frac{1}{x}\)B.\(\intf(x)dx=\lnx+C\)C.\(\int_{1}^{e}f(x)dx=1\)D.\(f(x)\)的不定積分是\(\lnx\)6.下列式子正確的是？A.\(\int\cos2xdx=\frac{1}{2}\sin2x+C\)B.\(\int\sin3xdx=-\frac{1}{3}\cos3x+C\)C.\(\inte^{3x}dx=\frac{1}{3}e^{3x}+C\)D.\(\int\frac{1}{x^2}dx=-\frac{1}{x}+C\)7.若\(\intf(x)dx=F(x)+C\)，則\(\intf(ax+b)dx=\)？A.\(F(ax+b)+C\)B.\(\frac{1}{a}F(ax+b)+C\)C.\(aF(ax+b)+C\)D.\(F(ax+b)\)8.下列積分值為\(0\)的有？A.\(\int_{-\pi}^{\pi}\sinxdx\)B.\(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\cosxdx\)C.\(\int_{-1}^{1}x^5dx\)D.\(\int_{-1}^{1}x^4dx\)9.已知\(f(x)\)滿(mǎn)足\(\int_{0}^{1}f(x)dx=1\)，則？A.\(f(x)\)在\([0,1]\)上恒為\(1\)B.\(\int_{0}^{1}2f(x)dx=2\)C.\(\int_{0}^{1}f(x)+1dx=2\)D.\(\int_{0}^{1}f(x)-1dx=0\)10.下列函數(shù)中，其導(dǎo)數(shù)等于\(\sin2x\)的有？A.\(-\frac{1}{2}\cos2x\)B.\(\frac{1}{2}\cos2x\)C.\(-\frac{1}{2}\cos2x+C\)D.\(\frac{1}{2}\cos2x+C\)答案：1.ABC2.ABCD3.AB4.AC5.ABC6.ABCD7.B8.ABC9.BC10.AC判斷題（每題2分，共10題）1.函數(shù)\(f(x)\)的原函數(shù)一定是連續(xù)函數(shù)。（）2.\(\int_{a}^f(x)dx=-\int_^{a}f(x)dx\)。（）3.若\(F(x)\)是\(f(x)\)的原函數(shù)，則\(F(x)+C\)也是\(f(x)\)的原函數(shù)。（）4.\(\int_{0}^{1}x^3dx=\frac{1}{4}\)。（）5.\(\int\frac{1}{x}dx=\lnx\)。（）6.\(\int_{-1}^{1}x^2\sinxdx=\int_{-1}^{1}x^2dx\cdot\int_{-1}^{1}\sinxdx\)。（）7.若\(f(x)\)在\([a,b]\)上可積，則\(f(x)\)在\([a,b]\)上一定連續(xù)。（）8.\(\int_{0}^{2\pi}\sinxdx=0\)。（）9.已知\(F^\prime(x)=f(x)\)，則\(\int_{a}^f(x)dx=F(b)-F(a)\)。（）10.\(\int_{0}^{1}e^xdx=e\)。（）答案：1.×2.√3.√4.√5.×6.×7.×8.√9.√10.×簡(jiǎn)答題（總4題，每題5分）1.簡(jiǎn)述原函數(shù)與不定積分的關(guān)系。原函數(shù)是一個(gè)函數(shù)，不定積分是原函數(shù)的集合。若\(F^\prime(x)=f(x)\)，則\(F(x)\)是\(f(x)\)的原函數(shù)，\(\intf(x)dx=F(x)+C\)。2.計(jì)算\(\int_{0}^{2}(2x+1)dx\)。\(\int_{0}^{2}(2x+1)dx=[x^2+x]_0^2=(2^2+2)-(0+0)=6\)。3.已知\(f(x)\)的一個(gè)原函數(shù)是\(x^3\)，求\(f(x)\)及\(\intf(x)dx\)。\(f(x)=3x^2\)，\(\intf(x)dx=x^3+C\)。4.計(jì)算\(\int_{1}^{2}\frac{1}{x^2}dx\)。\(\int_{1}^{2}\frac{1}{x^2}dx=[-\frac{1}{x}]_1^2=-\frac{1}{2}-(-1)=\frac{1}{2}\)。討論題（總4題，每題5分）1.討論\(\int_{-a}^{a}f(x)dx\)與\(f(x)\)奇偶性的關(guān)系。若\(f(x)\)為偶函數(shù)，\(\int_{-a}^{a}f(x)dx=2\int_{0}^{a}f(x)dx\)；若\(f(x)\)為奇函數(shù)，\(\int_{-a}^{a}f(x)dx=0\)。2.如何利用積分求曲線圍成圖形的面積？通過(guò)確定積分區(qū)間，找到被積函數(shù)，利用定積分計(jì)算\(\int_{a}^|f(x)-g