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

= 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

\int_{0}^{2} (2 x - x^{2}) d x

x \to 1 + lim (\frac{x - 1}{x + 1})

\frac{3 s - 1}{s ^{2} - 8 s + 21}

\int - e^{- x^{2}} d x

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

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