What is the integral of (5x^2-6)^{3/2} ?

The integral of (5x^2-6)^{3/2} is

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integral of (5x^2-6)^{3/2}

\int (5 x^{2} - 6)^{\frac{3}{2}} d x

x (5 x^{2} - 6)^{\frac{3}{2}} - 15 \cdot 6^{\frac{1}{2}} (\frac{6}{5})^{\frac{3}{2}} (- \frac{1}{48} (5^{\frac{1}{2}} x (5 x^{2} - 6)^{\frac{1}{2}} + 6 ln (\frac{5 x ^{2} - 6}{6})^{\frac{1}{2}} + (\frac{5}{6})^{\frac{1}{2}} x) + \frac{5 \cdot 5 ^{\frac{1}{2}}}{144} x^{3} (5 x^{2} - 6)^{\frac{1}{2}}) + C

Solution steps

\int (5 x^{2} - 6)^{\frac{3}{2}} d x

Apply Integration By Parts

: x (5 x^{2} - 6)^{\frac{3}{2}} - \int 15 x^{2} (5 x^{2} - 6)^{\frac{1}{2}} d x

= x (5 x^{2} - 6)^{\frac{3}{2}} - \int 15 x^{2} (5 x^{2} - 6)^{\frac{1}{2}} d x

\int 15 x^{2} (5 x^{2} - 6)^{\frac{1}{2}} d x = 15 \cdot 6^{\frac{1}{2}} (\frac{6}{5})^{\frac{3}{2}} (- \frac{1}{48} (5^{\frac{1}{2}} x (5 x^{2} - 6)^{\frac{1}{2}} + 6 ln (\frac{5 x ^{2} - 6}{6})^{\frac{1}{2}} + (\frac{5}{6})^{\frac{1}{2}} x) + \frac{5 \cdot 5 ^{\frac{1}{2}}}{144} x^{3} (5 x^{2} - 6)^{\frac{1}{2}})

= x (5 x^{2} - 6)^{\frac{3}{2}} - 15 \cdot 6^{\frac{1}{2}} (\frac{6}{5})^{\frac{3}{2}} (- \frac{1}{48} (5^{\frac{1}{2}} x (5 x^{2} - 6)^{\frac{1}{2}} + 6 ln (\frac{5 x ^{2} - 6}{6})^{\frac{1}{2}} + (\frac{5}{6})^{\frac{1}{2}} x) + \frac{5 \cdot 5 ^{\frac{1}{2}}}{144} x^{3} (5 x^{2} - 6)^{\frac{1}{2}})

Add a constant to the solution

= x (5 x^{2} - 6)^{\frac{3}{2}} - 15 \cdot 6^{\frac{1}{2}} (\frac{6}{5})^{\frac{3}{2}} (- \frac{1}{48} (5^{\frac{1}{2}} x (5 x^{2} - 6)^{\frac{1}{2}} + 6 ln (\frac{5 x ^{2} - 6}{6})^{\frac{1}{2}} + (\frac{5}{6})^{\frac{1}{2}} x) + \frac{5 \cdot 5 ^{\frac{1}{2}}}{144} x^{3} (5 x^{2} - 6)^{\frac{1}{2}}) + C

\frac{d}{d x} (\frac{12}{ln ( x )})

\int \frac{8 t ^{3} - 3 t ^{2} + 8 t}{t ^{2} + 1} d t

\int - 0.09 t d t

\frac{d}{d x} (e^{- 5 x^{3} - 2 x^{2}})

x \to 4 + lim (\frac{1}{( x - 4 ) ^{2}})