Question

Someone help !! Struggling with this one

Answer 1

Answer:

$2\sqrt{10}-4\sqrt{2}$

Step-by-step explanation:

Evaluate:

$\int\limits^7_5 {\dfrac{1}{\sqrt{3+t}}} \, dt$

Rewrite it as

$\int\limits^7_5 {\dfrac{1}{(3+t)^{\frac{1}{2}}}} \, dt=\int\limits^7_5 {(3+t)^{-\frac{1}{2}}} \, dt$

Use rule

$\int\limits^a_b {(x+k)^n} \, dx=\dfrac{(b+k)^{n+1}}{n+1}-\dfrac{(a+k)^{n+1}}{n+1}$

Hence,

$\int\limits^7_5 {(3+t)^{-\frac{1}{2}}} \, dt=\dfrac{(7+3)^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}-\dfrac{(5+3)^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}=\\ \\=\dfrac{10^{\frac{1}{2}}}{\frac{1}{2}}-\dfrac{8^{\frac{1}{2}}}{\frac{1}{2}}=2\sqrt{10}-2\sqrt{8}=2\sqrt{10}-4\sqrt{2}$