What is the integral of (3x+4)^{5/2} ?

The integral of (3x+4)^{5/2} is 2/7 x(3x+4)^{5/2}+8/21 (3x+4)^{5/2}+C

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integral of (3x+4)^{5/2}

\int (3 x + 4)^{\frac{5}{2}} d x

\frac{2}{7} x (3 x + 4)^{\frac{5}{2}} + \frac{8}{21} (3 x + 4)^{\frac{5}{2}} + C

Solution steps

\int (3 x + 4)^{\frac{5}{2}} d x

Apply Integration By Parts

= x (3 x + 4)^{\frac{5}{2}} - \int \frac{15}{2} x (3 x + 4)^{\frac{3}{2}} d x

\int \frac{15}{2} x (3 x + 4)^{\frac{3}{2}} d x = \frac{1}{21} (15 x - 8) (3 x + 4)^{\frac{5}{2}}

= x (3 x + 4)^{\frac{5}{2}} - \frac{1}{21} (15 x - 8) (3 x + 4)^{\frac{5}{2}}

Simplify

x (3 x + 4)^{\frac{5}{2}} - \frac{1}{21} (15 x - 8) (3 x + 4)^{\frac{5}{2}} : \frac{2}{7} x (3 x + 4)^{\frac{5}{2}} + \frac{8}{21} (3 x + 4)^{\frac{5}{2}}

= \frac{2}{7} x (3 x + 4)^{\frac{5}{2}} + \frac{8}{21} (3 x + 4)^{\frac{5}{2}}

Add a constant to the solution

= \frac{2}{7} x (3 x + 4)^{\frac{5}{2}} + \frac{8}{21} (3 x + 4)^{\frac{5}{2}} + C

d er i v a t i v e 1 + x

x \to 0 + lim (x^{2} e^{\frac{1}{x}})

\frac{d}{d x} (2 e^{- 2 x} + \frac{1}{3} e^{x})

\frac{d}{d x} (6^{x^{2} - 3})

\frac{d}{d x} (\frac{x - x}{x ^{2}})

What is the integral of (3x+4)^{5/2} ?
The integral of (3x+4)^{5/2} is 2/7 x(3x+4)^{5/2}+8/21 (3x+4)^{5/2}+C