Question

Someone help !! Struggling with this one

Answer 1

`2√(10)-4√(2)`

Explanation:

Evaluate:

`\int\limits^7_5 {(1)/(√(3+t))} \, dt`

Rewrite it as

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

Use rule

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

Hence,

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