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

The integral of 3x^2sqrt(x^3)+1 is 2/3 x^{9/2}+x+C

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

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

\frac{2}{3} x^{\frac{9}{2}} + x + C

Solution steps

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

x^{3} = x^{\frac{3}{2}},

assuming

x \geq 0

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

\int 3 x^{2} x^{\frac{3}{2}} d x = \frac{2}{3} x^{\frac{9}{2}}

\int 1 d x = x

= \frac{2}{3} x^{\frac{9}{2}} + x

Add a constant to the solution

= \frac{2}{3} x^{\frac{9}{2}} + x + C

\int \frac{9 x ^{3}}{1 + x ^{2}} d x

\frac{d}{d x} (e^{- 2 x} \cdot cos (x))

x \to 0 + lim (3 x^{(\frac{x}{2})})

\int \frac{2 x + 1}{( x + 1 ) ^{3} ( x ^{2} + 4 ) ^{2}} d x

\int x - 8 d x

What is the integral of 3x^2sqrt(x^3)+1 ?
The integral of 3x^2sqrt(x^3)+1 is 2/3 x^{9/2}+x+C