Integral Example - # 7

[Jason76]

Evaluated at top 16 and bottom 1

\(\displaystyle \int \dfrac{x - 7}{\sqrt{x}}\)

\(\displaystyle \int \dfrac{x - 7}{x^{1/2}}\)

\(\displaystyle \int \dfrac{x}{x^{1/2}} - \dfrac{7}{x^{1/2}} \)

\(\displaystyle \int x^{1/2} - \dfrac{7}{x^{1/2}}\)

\(\displaystyle \int x^{1/2} -7x^{-1/2}\)

\(\displaystyle \rightarrow \dfrac{x^{3/2}}{\dfrac{3}{2}} - \dfrac{7x^{1/2}}{\dfrac{1}{2}}\)

\(\displaystyle \rightarrow (\dfrac{2}{3})x^{3/2} - (\dfrac{2}{1})7x^{1/2}\)

\(\displaystyle \rightarrow \dfrac{2}{3}x^{3/2} - 14x^{1/2}\)

\(\displaystyle [\dfrac{2}{3}(16)^{3/2} - 14(16)^{1/2}] - [\dfrac{2}{3}(1)^{3/2} - 14(1)^{1/2}] \)

\(\displaystyle [\dfrac{2}{3}(64)- 14(4)] - [\dfrac{2}{3}(1) - 14(1)] \)

\(\displaystyle [\dfrac{128}{3} - 56] - [\dfrac{2}{3} - 14] \)

 

[fcabanski]

Now plug in 1 and 16 to determine the definite integral.

You can verify your answer on wolframalpha.com. Your integration is correct, but the answer isn't expressed in simplest terms.

Evaluated at top 16 and bottom 1

\(\displaystyle \int \dfrac{x - 7}{\sqrt{x}}\)

\(\displaystyle \int \dfrac{x - 7}{x^{1/2}}\)

\(\displaystyle \int \dfrac{x}{x^{1/2}} - \dfrac{7}{x^{1/2}} \)

\(\displaystyle \int x^{1/2} - \dfrac{7}{x^{1/2}}\)

\(\displaystyle \int x^{1/2} -7x^{-1/2}\)

\(\displaystyle \rightarrow \dfrac{x^{3/2}}{\dfrac{3}{2}} - \dfrac{7x^{1/2}}{\dfrac{1}{2}}\)

\(\displaystyle \rightarrow (\dfrac{2}{3})x^{3/2} - (\dfrac{2}{1})7x^{1/2}\)

\(\displaystyle \rightarrow \dfrac{2}{3}x^{3/2} - 14x^{1/2}\)

\(\displaystyle [\dfrac{2}{3}(16)^{3/2} - 14(16)^{1/2}] - [\dfrac{2}{3}(1)^{3/2} - 14(1)^{1/2}] \)

 

[Jason76]

Evaluated at top 16 and bottom 1

\(\displaystyle \int \dfrac{x - 7}{\sqrt{x}}\)

\(\displaystyle \int \dfrac{x - 7}{x^{1/2}}\)

\(\displaystyle \int \dfrac{x}{x^{1/2}} - \dfrac{7}{x^{1/2}} \)

\(\displaystyle \int x^{1/2} - \dfrac{7}{x^{1/2}}\)

\(\displaystyle \int x^{1/2} -7x^{-1/2}\)

\(\displaystyle \rightarrow \dfrac{x^{3/2}}{\dfrac{3}{2}} - \dfrac{7x^{1/2}}{\dfrac{1}{2}}\)

\(\displaystyle \rightarrow (\dfrac{2}{3})x^{3/2} - (\dfrac{2}{1})7x^{1/2}\)

\(\displaystyle \rightarrow \dfrac{2}{3}x^{3/2} - 14x^{1/2}\)

\(\displaystyle [\dfrac{2}{3}(16)^{3/2} - 14(16)^{1/2}] - [\dfrac{2}{3}(1)^{3/2} - 14(1)^{1/2}] \)

\(\displaystyle [\dfrac{2}{3}(64)- 14(4)] - [\dfrac{2}{3}(1) - 14(1)] \)

\(\displaystyle [\dfrac{128}{3} - 56] - [\dfrac{2}{3} - 14] \)

\(\displaystyle [\dfrac{128}{3} - ]dfrac{168}{3}] - [\dfrac{2}{3} - \dfrac{42}{3}] \)

\(\displaystyle [-\dfrac{40}{3}] - [-\dfrac{40}{3}]\)

\(\displaystyle [-\dfrac{40}{3}] + \dfrac{40}{3}] = 0\) - This is answer IS correct on computer. No need to go further.