Answer:
[tex]2\sqrt{10}-4\sqrt{2}[/tex]
Step-by-step explanation:
Evaluate:
[tex]\int\limits^7_5 {\dfrac{1}{\sqrt{3+t}}} \, dt[/tex]
Rewrite it as
[tex]\int\limits^7_5 {\dfrac{1}{(3+t)^{\frac{1}{2}}}} \, dt=\int\limits^7_5 {(3+t)^{-\frac{1}{2}}} \, dt[/tex]
Use rule
[tex]\int\limits^a_b {(x+k)^n} \, dx=\dfrac{(b+k)^{n+1}}{n+1}-\dfrac{(a+k)^{n+1}}{n+1}[/tex]
Hence,
[tex]\int\limits^7_5 {(3+t)^{-\frac{1}{2}}} \, dt=\dfrac{(7+3)^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}-\dfrac{(5+3)^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}=\\ \\=\dfrac{10^{\frac{1}{2}}}{\frac{1}{2}}-\dfrac{8^{\frac{1}{2}}}{\frac{1}{2}}=2\sqrt{10}-2\sqrt{8}=2\sqrt{10}-4\sqrt{2}[/tex]
