Chapter 2: Integral Calculus I January 7, 2025 Maven Leave a Comment Welcome to your Chapter 2: Integral Calculus I 1. $\displaystyle \int_{}^{}\frac{e^{x}}{e^{x}+1}\; dx\; is$ $log\left| \frac{e^{x}}{e^{x}+1} \right|+c$ $log\left| \frac{e^{x}+1}{e^{x}} \right|+c$ $log\left| e^{x} \right|+c$ $log\left| e^{x} +1\right|+c$ None 2. $\displaystyle \int_{}^{}\frac{sin 5x-sinx}{cos3x} \;dx \;is $ $−cos2x + c$ $−cos2x + c$ $\frac{-1}{4} cos2x c$ $−4cos2x + c$ None 3. $\displaystyle \int_{}^{}\sqrt{e^{x}}\;dx\; is$ $\sqrt{e^{x}}+c$ $2\sqrt{e^{x}}+c$ $\frac{1}{2}\sqrt{e^{x}}+c$ $\frac{1}{2\sqrt{e^{x}}}+c$ None 4. $\displaystyle \int_{}^{}\frac{1}{x^{3}} dx$ $\frac{-3}{x^{2}} +c$ $\frac{-1}{2x^{2}} +c$ $\frac{-1}{3x^{2}} +c$ $\frac{-2}{x^{2}} +c$ None 5. $\displaystyle \int_{}^{}e^{2x}[2x^{2}+2x]\; dx$ $e^{2x}x^{2}+c$ $xe^{2x}+c$ $2x^{2}e^{2}+c$ $\frac{x^{2}e^{x}}{2}+c$ None 6. $\displaystyle \int_{}^{}\dfrac{{e^{x}}}{\sqrt{1+e^{x}}}\; dx \; is$ $\dfrac{{e^{x}}}{\sqrt{1+e^{x}}}+c$ $2\sqrt{1+e^{x}}+c$ $\sqrt{1+e^{x}}+c$ $e^{x}\sqrt{1+e^{x}}+c$ None 7. $\displaystyle \int_{}^{} \bigg[\frac{9}{x-3}-\frac{1}{x+1}\bigg ]\; dx \; is$ $log \left| x − 3 \right| − log \left| x +1 \right| + c$ $log \left| x − 3 \right| + log \left| x +1 \right| + c$ $9log \left| x − 3 \right| − log \left| x +1 \right| + c$ $9log \left| x − 3 \right| + log \left| x +1 \right| + c$ None 8. $\displaystyle \int_{}^{}\frac{logx}{x}\; dx, \; x\gt 0\; is $ $\frac{1}{2}\;(logx )^{2}\;+c$ $\frac{-1}{2}\;(logx )^{2}$ $\frac{2}{x^{2}}+c$ None 9. $\displaystyle \int_{}^{}\frac{sin 2x}{2 sin x} dx is $ $ sin x + c $ $\frac{1}{2}sin x + c$ $cos x + c$ $\frac{1}{2}cos x + c$ None 10. $\displaystyle \int_{}^{}2^{x} dx$ $2^{x} log\;2 + c$ $2^{x} + c$ $\dfrac{2^{x}}{log \;2}+c$ $\dfrac{log\;2}{2^{x}}+c$ None 11. $\displaystyle \int_{0}^{\frac{\pi}{3}}tanx\;dx\;is$ $log2$ $0$ $log\sqrt{2}$ $2log2$ None 12. $The\;value\;of\;\displaystyle \int_{2}^{3}f(5-x)\;dx-\int_{2}^{3}f(x)\;dx\;is$ $1$ $0$ $-1$ $5$ None 13. $Using\; the \;factorial \;representation\; of \;the\; gamma \;function,\;which \;of\; the \;following \;is\; the\; solution \\ \;for \;the \;gamma \;function\;\Gamma(n) \;when \;n = 8$ $5040$ $5400$ $4500$ $5540$ None 14. $\Gamma\;1\;is$ $0$ $1$ $n$ $n!$ None 15. $If\; n\gt 0, \;then\;\Gamma\;(n)\; is$ $\int_{0}^{1}e^{-x}x^{n-1}\;dx$ $\int_{0}^{1}e^{-x}x^{n}\;dx$ $\int_{0}^{\infty }e^{x}x^{-n}\;dx$ $\int_{0}^{\infty }e^{-x}x^{n-1}\;dx$ None 16. $If\; \displaystyle \int_{0}^{1}f(x)\;dx\;=1, \; \displaystyle \int_{0}^{1}x\;f(x)\;dx\;=a\; and \int_{0}^{1}(x^{2}f(x)\;dx\;=a^{2},then\; \int_{0}^{1}(a-x)^{2}f(x)\;dx\;is$ $4a^{2}$ $0$ $2a^{2}$ $1$ None 17. $\displaystyle \int_{0}^{\infty }x^{4}e^{-x}\;dx\;is$ $12$ $4$ $4!$ $64$ None 18. $\Gamma \frac{3}{2}$ $\sqrt{\pi}$ $\frac{\sqrt{\pi}}{2}$ $2\sqrt{\pi}$ $\frac{3}{2}$ None 19. $\Gamma(n) \;is$ $(n-1)!$ $n!$ $n\Gamma(n)$ $(n-1)\Gamma(n)$ None 20. $\displaystyle \int_{0}^{4}\bigg(\sqrt{x}+\frac{1}{\sqrt{x}}\bigg) \;dx\;is$ $\frac{20}{3}$ $\frac{21}{3}$ $\frac{28}{3}$ $\frac{1}{3}$ None 21. $\displaystyle \int_{2}^{4}\frac{dx}{x}\; is$ $log 4$ $0$ $log 2$ $log 8$ None 22. $\displaystyle \int_{}^{}\frac{dx}{\sqrt{x^{2}-36}}\; is$ $\sqrt{x^{2}-36}+c$ $log\left| x+\sqrt{x^{2}-36} \right|+c$ $log\left| x-\sqrt{x^{2}-36} \right|+c$ $log\left| x^{2}+\sqrt{x^{2}-36} \right|+c$ None 23. $\displaystyle \int_{0}^{\infty }e^{-2x}\;dx\; is$ $0$ $1$ $2$ $\frac{1}{2}$ None 24. $The \; value \; of\; \displaystyle \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}cosx\;dx\; is$ $0$ $2$ $1$ $4$ None 25. $\displaystyle \int_{}^{}\frac{2x^{3}}{4+x^{4}}\; dx\; is$ $log \left| 4+x^{4} \right|+c$ $\frac{1}{2}log \left| 4+x^{4} \right|+c$ $\frac{1}{4}log \left| 4+x^{4} \right|+c$ $log \left| \frac{2x^{3}}{4+x^{4}} \right|+c$ None 26. $\displaystyle \int_{_0}^{_1}\sqrt{x^{4}(1-x)^{2}}\;dx\;is$ $\frac{1}{12}$ $\frac{-7}{12}$ $\frac{7}{12}$ $\frac{-1}{12}$ None 27. $\displaystyle \int_{}^{}\frac{2x+3}{\sqrt{x^{2}+3x+2}}\; dx\; is$ $\sqrt{x^{2}+3x+2}+c$ $2 \sqrt{x^{2}+3x+2}+c$ $log({x^{2}+3x+2})+c$ $\frac{2}{3}({x^{2}+3x+2})^{\frac{3}{2}}+c$ None 28. $\displaystyle \int_{-1}^{1}x^{3}e^{x^{4}}\; dx\; is$ $1$ $2\int_{0}^{1}x^{3}e^{x^{4}}\; dx$ $0$ $e^{x^{4}}$ None 29. $If\; f (x)\; is\; a\; continuous\; function\; and\;a\lt c\lt b, \;then,\; \displaystyle \int_{a}^{c}f(x)\;dx+ \int_{c}^{b}f(x)\;dx\; is$ $\int_{a}^{b}f(x)\;dx- \int_{a}^{c}f(x)\;dx$ $\int_{a}^{c}f(x)\;dx- \int_{a}^{b}f(x)\;dx$ $\int_{a}^{b}f(x)\;dx$ $0$ None 30. $\displaystyle \int_{0}^{1}(2x+1)\; dx\; is$ $1$ $2$ $3$ $4$ None Time's up Related Posts:Chapter 3: Integral Calculus IIChapter 5: Differential CalculusChapter 1: Introduction to Micro EconomicsChapter 2. Consumption Analysis
Leave a Reply