What is the integral of x^{1/2}+1/(x+7) ?

The integral of x^{1/2}+1/(x+7) is 2/3 x^{3/2}+ln|x+7|+C

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integral of x^{1/2}+1/(x+7)

\int x^{\frac{1}{2}} + \frac{1}{x + 7} d x

\frac{2}{3} x^{\frac{3}{2}} + ln ∣ x + 7 ∣ + C

Solution steps

\int x^{\frac{1}{2}} + \frac{1}{x + 7} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x^{\frac{1}{2}} d x + \int \frac{1}{x + 7} d x

\int x^{\frac{1}{2}} d x = \frac{2}{3} x^{\frac{3}{2}}

\int \frac{1}{x + 7} d x = ln ∣ x + 7 ∣

= \frac{2}{3} x^{\frac{3}{2}} + ln ∣ x + 7 ∣

Add a constant to the solution

= \frac{2}{3} x^{\frac{3}{2}} + ln ∣ x + 7 ∣ + C

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What is the integral of x^{1/2}+1/(x+7) ?
The integral of x^{1/2}+1/(x+7) is 2/3 x^{3/2}+ln|x+7|+C