1. ∫[tex]3x^{2}\sqrt{x^{3}+77}[/tex] [tex]dx[/tex]
[tex]anggap[/tex] [tex]u=x^{3} +77[/tex]
[tex]du=3x^{2}[/tex] [tex]dx[/tex]
[tex]dx=\frac{du}{3x^{2} }[/tex]
[tex]3[/tex]∫[tex]x^{2} \sqrt{u}[/tex] [tex]du[/tex]
[tex]3[/tex]∫[tex]x^{2} \sqrt{u}[/tex] [tex]\frac{du}{3x^{2} }[/tex]
[tex]3.\frac{1}{3}[/tex]∫[tex]\sqrt{u}[/tex] [tex]du[/tex]
∫[tex]u^{\frac{1}{2} }[/tex] [tex]du[/tex] = [tex]\frac{1}{\frac{3}{2} } u^{\frac{3}{2} }[/tex]
[tex]\frac{2}{3} u^{\frac{3}{2} }[/tex] = [tex]\frac{2}{3} (x^{3}+77 )^{\frac{3}{2} } +C[/tex]
2. ∫[tex]2x(x^{2} +5)^{5}[/tex] [tex]dx[/tex]
[tex]anggap[/tex] [tex]u=x^{2} +5[/tex]
[tex]du=2x[/tex] [tex]dx[/tex]
[tex]dx=\frac{du}{2x}[/tex]
[tex]2[/tex]∫[tex]x.u^{5}[/tex] [tex]du[/tex]
[tex]2[/tex]∫[tex]x.u^{5}[/tex] [tex]\frac{du}{2x}[/tex]
[tex]2.\frac{1}{2}[/tex]∫[tex]u^{5}[/tex] [tex]du[/tex]
∫[tex]u^{5[/tex] [tex]du[/tex] = [tex]\frac{1}{6} u^{6}[/tex]
[tex]\frac{1}{6} (x^{2}+5)^{6} +C[/tex]
3. ∫[tex]cos(5x-1)dx[/tex]
[tex]anggap[/tex] [tex]u=5x-1[/tex]
[tex]du=5 dx\\ dx=\frac{du}{5}[/tex]
∫[tex]cos[/tex] [tex]u[/tex] [tex]dx[/tex]
∫[tex]cos[/tex] [tex]u.\frac{du}{5}[/tex]
[tex]\frac{1}{5}[/tex]∫[tex]cos[/tex] [tex]u[/tex] [tex]du[/tex] = [tex]\frac{1}{5}sin[/tex] [tex]u[/tex]
[tex]\frac{1}{5} sin (5x-1)+C[/tex]
4.
[tex]\int\limits^1_2 {(1+\frac{1}{t} )^{2} } \frac{1}{t^{2} } \, dt \\-\int\limits^2_1 {(1+\frac{2}{t}+\frac{1}{t^{2} } )\frac{1}{t^{2} } } \, dt \\-\int\limits^2_1 {(\frac{1}{t^{2} }+\frac{2}{t^{3} } +\frac{1}{t^{4} } )} \, dt \\-[(\int\limits^2_1 {\frac{1}{t^{2} } } \, dt )+(\int\limits^2_1 {\frac{2}{t^{3} } } \, dt )+(\int\limits^2_1 {\frac{1}{t^{4} } } \, dt )][/tex]
[tex]\int\limits^2_1 {\frac{1}{t^{2} } } \, dt = (\frac{-1}{2})-(\frac{-1}{1}) = \frac{1}{2}[/tex]
[tex]\int\limits^2_1 {\frac{2}{t^{3} } } \, dt = (\frac{-1}{4} )-(\frac{-1}{1} ) = \frac{3}{4}[/tex]
[tex]\int\limits^2_1 {\frac{1}{t^{4} } } \, dt = (\frac{-1}{24} ) - (\frac{-1}{3} ) = \frac{7}{24}[/tex]
[tex]-(\frac{1}{2} +\frac{3}{4} +\frac{7}{24} ) = - \frac{37}{24}[/tex]
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1. ∫[tex]3x^{2}\sqrt{x^{3}+77}[/tex] [tex]dx[/tex]
[tex]anggap[/tex] [tex]u=x^{3} +77[/tex]
[tex]du=3x^{2}[/tex] [tex]dx[/tex]
[tex]dx=\frac{du}{3x^{2} }[/tex]
[tex]3[/tex]∫[tex]x^{2} \sqrt{u}[/tex] [tex]du[/tex]
[tex]3[/tex]∫[tex]x^{2} \sqrt{u}[/tex] [tex]\frac{du}{3x^{2} }[/tex]
[tex]3.\frac{1}{3}[/tex]∫[tex]\sqrt{u}[/tex] [tex]du[/tex]
∫[tex]u^{\frac{1}{2} }[/tex] [tex]du[/tex] = [tex]\frac{1}{\frac{3}{2} } u^{\frac{3}{2} }[/tex]
[tex]\frac{2}{3} u^{\frac{3}{2} }[/tex] = [tex]\frac{2}{3} (x^{3}+77 )^{\frac{3}{2} } +C[/tex]
2. ∫[tex]2x(x^{2} +5)^{5}[/tex] [tex]dx[/tex]
[tex]anggap[/tex] [tex]u=x^{2} +5[/tex]
[tex]du=2x[/tex] [tex]dx[/tex]
[tex]dx=\frac{du}{2x}[/tex]
[tex]2[/tex]∫[tex]x.u^{5}[/tex] [tex]du[/tex]
[tex]2[/tex]∫[tex]x.u^{5}[/tex] [tex]\frac{du}{2x}[/tex]
[tex]2.\frac{1}{2}[/tex]∫[tex]u^{5}[/tex] [tex]du[/tex]
∫[tex]u^{5[/tex] [tex]du[/tex] = [tex]\frac{1}{6} u^{6}[/tex]
[tex]\frac{1}{6} (x^{2}+5)^{6} +C[/tex]
3. ∫[tex]cos(5x-1)dx[/tex]
[tex]anggap[/tex] [tex]u=5x-1[/tex]
[tex]du=5 dx\\ dx=\frac{du}{5}[/tex]
∫[tex]cos[/tex] [tex]u[/tex] [tex]dx[/tex]
∫[tex]cos[/tex] [tex]u.\frac{du}{5}[/tex]
[tex]\frac{1}{5}[/tex]∫[tex]cos[/tex] [tex]u[/tex] [tex]du[/tex] = [tex]\frac{1}{5}sin[/tex] [tex]u[/tex]
[tex]\frac{1}{5} sin (5x-1)+C[/tex]
4.
[tex]\int\limits^1_2 {(1+\frac{1}{t} )^{2} } \frac{1}{t^{2} } \, dt \\-\int\limits^2_1 {(1+\frac{2}{t}+\frac{1}{t^{2} } )\frac{1}{t^{2} } } \, dt \\-\int\limits^2_1 {(\frac{1}{t^{2} }+\frac{2}{t^{3} } +\frac{1}{t^{4} } )} \, dt \\-[(\int\limits^2_1 {\frac{1}{t^{2} } } \, dt )+(\int\limits^2_1 {\frac{2}{t^{3} } } \, dt )+(\int\limits^2_1 {\frac{1}{t^{4} } } \, dt )][/tex]
[tex]\int\limits^2_1 {\frac{1}{t^{2} } } \, dt = (\frac{-1}{2})-(\frac{-1}{1}) = \frac{1}{2}[/tex]
[tex]\int\limits^2_1 {\frac{2}{t^{3} } } \, dt = (\frac{-1}{4} )-(\frac{-1}{1} ) = \frac{3}{4}[/tex]
[tex]\int\limits^2_1 {\frac{1}{t^{4} } } \, dt = (\frac{-1}{24} ) - (\frac{-1}{3} ) = \frac{7}{24}[/tex]
[tex]-(\frac{1}{2} +\frac{3}{4} +\frac{7}{24} ) = - \frac{37}{24}[/tex]