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Section 6.8 Additional Exercise
These problems require the techniques of this chapter, and are in no particular order. Some problems may be done in more than one way.
Exercises Exercises
1.
\(\ds\int (t+4)^3\,dt\)
2.
\(\ds\int t(t^2-9)^{3/2}\,dt\)
Answer.\(\ds{(t^2-9)^{5/2}\over5}+C\)
3.
\(\ds\int (e^{t^2}+16)te^{t^2}\,dt\)
Answer.\(\ds{(e^{t^2}+16)^2\over 4}+C\)
4.
\(\ds\int \sin t\cos 2t\,dt\)
Answer.\(\ds\cos t-{2\over3}\cos^3 t+C\)
5.
\(\ds\int \tan t\sec^2t\,dt\)
6.
\(\ds\int {2t+1\over t^2+t+3}\,dt\)
7.
\(\ds\int {1\over t(t^2-4)}\,dt\)
Answer.\(\ds {1\over8} \ln|1-4/t^2|+C\)
8.
\(\ds\int {1\over (25-t^2)^{3/2}}\,dt\)
Answer.\(\ds{1\over25}\tan(\arcsin(t/5))+C={t\over25\sqrt{25-t^2}}+C\)
9.
\(\ds\int {\cos 3t\over\sqrt{\sin3t}}\,dt\)
Answer.\(\ds{2\over3}\sqrt{\sin 3t}+C\)
10.
\(\ds\int t\sec^2 t\,dt\)
Answer.\(\ds t\tan t+\ln|\cos t|+C\)
11.
\(\ds\int {e^t\over \sqrt{e^t+1}}\,dt\)
12.
\(\ds\int \cos^4 t\,dt\)
Answer.\(\ds{3t\over 8}+{\sin 2t\over4}+ {\sin 4t\over 32}+C\)
13.
\(\ds\int {1\over t^2+3t}\,dt\)
Answer.\(\ds{\ln |t|\over 3} - {\ln |t+3|\over 3}+C\)
14.
\(\ds\int {1\over t^2\sqrt{1+t^2}}\,dt\)
Answer.\(\ds{-1\over \sin\arctan t}+C=-\sqrt{1+t^2}/t+C\)
15.
\(\ds\int {\sec^2t\over (1+\tan t)^3}\,dt\)
Answer.\(\ds{-1\over 2(1+\tan t)^2}+C\)
16.
\(\ds\int t^3\sqrt{t^2+1}\,dt\)
Answer.\(\ds{(t^2+1)^{5/2}\over 5}-{(t^2+1)^{3/2}\over 3}+C\)
17.
\(\ds\int e^t\sin t\,dt\)
Answer.\(\ds{e^t\sin t-e^t\cos t\over 2}+C\)
18.
\(\ds\int (t^{3/2}+47)^3\sqrt{t}\,dt\)
Answer.\(\ds{(t^{3/2}+47)^4\over6}+C\)
19.
\(\ds\int {t^3\over (2-t^2)^{5/2}}\,dt\)
Answer.\(\ds{2\over 3(2-t^2)^{3/2}}-{1\over(2-t^2)^{1/2}}+C\)
20.
\(\ds\int {1\over t(9+4t^2)}\,dt\)
Answer.\(\ds{\ln|\sin(\arctan(2t/3))|\over9}+C =
{\ln(4t^2)-\ln(9+4t^2)\over 18} + C\)
21.
\(\ds\int {\arctan 2t\over 1+4t^2}\,dt\)
Answer.\(\ds{(\arctan(2t))^2\over4}+C\)
22.
\(\ds\int {t\over t^2+2t-3}\,dt\)
Answer.\(\ds{3\ln|t+3|\over 4}+{\ln|t-1|\over4}+C\)
23.
\(\ds\int \sin^3 t\cos^4 t\,dt\)
Answer.\(\ds{\cos^7 t\over 7}-{\cos^5 t\over 5}+C\)
24.
\(\ds\int {1\over t^2-6t+9}\,dt\)
25.
\(\ds\int {1\over t(\ln t)^2}\,dt\)
26.
\(\ds\int t(\ln t)^2\,dt\)
Answer.\(\ds{t^2(\ln t)^2\over 2}-{t^2\ln t\over 2}+{t^2\over4}+C\)
27.
\(\ds\int t^3e^{t}\,dt\)
28.
\(\ds\int {t+1\over t^2+t-1}\,dt\)
Answer.\(\ds{5+\sqrt5\over10}
\ln(2t+1-\sqrt5)+{5-\sqrt5\over10}\ln(2t+1+\sqrt5)+C\)
