Question

tính tích phân 

a) \(\int\limits^4_14\sqrt{x}dx\)

b) \(\int\limits^9_1\dfrac{5}{\sqrt{x}}dx\)

Answer 1

a: \(\int4\sqrt{x}dx=4\cdot\int x^{\dfrac{1}{2}}dx=4\cdot\dfrac{1}{\dfrac{1}{2}+1}\cdot x^{\dfrac{1}{2}+1}+C\)

\(=\dfrac{8}{3}x^{\dfrac{3}{2}}+C\)

\(\int_1^44\sqrt{x}dx=\left(\dfrac{8}{3}\cdot4^{\dfrac{3}{2}}+C\right)-\left(\dfrac{8}{3}\cdot1^{\dfrac{3}{2}}+C\right)=\dfrac{56}{3}\)

b: \(\int\dfrac{5}{\sqrt{x}}dx=\int5\cdot x^{-\dfrac{1}{2}}dx=5\cdot\dfrac{1}{-\dfrac{1}{2}+1}\cdot x^{-\dfrac{1}{2}+1}+C\)

\(=10\cdot x^{\dfrac{1}{2}}+C\)

\(\int_1^9\dfrac{5}{\sqrt{x}}dx=10\cdot9^{\dfrac{1}{2}}+C-\left(10\cdot1^{\dfrac{1}{2}}+C\right)=10\cdot3-10\cdot1=20\)