What is the integral of x^3+1/x+sqrt(1+2x) ?

The integral of x^3+1/x+sqrt(1+2x) is (x^4)/4+ln|x|+1/3 (1+2x)^{3/2}+C

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integral of x^3+1/x+sqrt(1+2x)

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

\frac{x ^{4}}{4} + ln ∣ x ∣ + \frac{1}{3} (1 + 2 x)^{\frac{3}{2}} + C

Solution steps

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

\int x^{3} d x = \frac{x ^{4}}{4}

\int \frac{1}{x} d x = ln ∣ x ∣

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

= \frac{x ^{4}}{4} + ln ∣ x ∣ + \frac{1}{3} (1 + 2 x)^{\frac{3}{2}}

Add a constant to the solution

= \frac{x ^{4}}{4} + ln ∣ x ∣ + \frac{1}{3} (1 + 2 x)^{\frac{3}{2}} + C

What is the integral of x^3+1/x+sqrt(1+2x) ?
The integral of x^3+1/x+sqrt(1+2x) is (x^4)/4+ln|x|+1/3 (1+2x)^{3/2}+C