What is the integral of [sqrt(x)(x^2-1)+csc^2(x)] ?

The integral of [sqrt(x)(x^2-1)+csc^2(x)] is 2/7 x^{7/2}-2/3 x^{3/2}-cot(x)+C

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integral of [sqrt(x)(x^2-1)+csc^2(x)]

\int [x (x^{2} - 1) + csc^{2} (x)] d x

\frac{2}{7} x^{\frac{7}{2}} - \frac{2}{3} x^{\frac{3}{2}} - cot (x) + C

Solution steps

\int x (x^{2} - 1) + csc^{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 (x^{2} - 1) d x + \int csc^{2} (x) d x

\int x (x^{2} - 1) d x = \frac{2}{7} x^{\frac{7}{2}} - \frac{2}{3} x^{\frac{3}{2}}

\int csc^{2} (x) d x = - cot (x)

= \frac{2}{7} x^{\frac{7}{2}} - \frac{2}{3} x^{\frac{3}{2}} - cot (x)

Add a constant to the solution

= \frac{2}{7} x^{\frac{7}{2}} - \frac{2}{3} x^{\frac{3}{2}} - cot (x) + C

4 sin (x) sin (x)

\frac{s + 5}{s ^{2} + 5 s + 9}

y^{^{''}} + 4 y = t^{2} + 3 e^{t}, y (0) = 0, y^{^{'}} (0) = 2

\int y^{3} d y

y = (\frac{( x - 1 )}{( x + 1 )})^{2}, at x = 0

What is the integral of [sqrt(x)(x^2-1)+csc^2(x)] ?
The integral of [sqrt(x)(x^2-1)+csc^2(x)] is 2/7 x^{7/2}-2/3 x^{3/2}-cot(x)+C