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

The integral of (1+sin^2(x))/(sqrt(x)) is 3sqrt(x)-(sqrt(pi))/2 \C(sqrt(2)sqrt(2/pi)sqrt(x))+C

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integral of (1+sin^2(x))/(sqrt(x))

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

3 x - \frac{π}{2} C (2 \frac{2}{π} x) + C

Solution steps

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

Expand

\frac{1 + sin ^{2} ( x )}{x} : \frac{1}{x} + \frac{sin ^{2} ( x )}{x}

= \int \frac{1}{x} + \frac{sin ^{2} ( x )}{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 \frac{1}{x} d x + \int \frac{sin ^{2} ( x )}{x} d x

\int \frac{1}{x} d x = 2 x

\int \frac{sin ^{2} ( x )}{x} d x = \frac{1}{2} (2 x - π C (2 \frac{2}{π} x))

= 2 x + \frac{1}{2} (2 x - π C (2 \frac{2}{π} x))

Simplify

2 x + \frac{1}{2} (2 x - π C (2 \frac{2}{π} x)) : 3 x - \frac{π}{2} C (2 \frac{2}{π} x)

= 3 x - \frac{π}{2} C (2 \frac{2}{π} x)

Add a constant to the solution

= 3 x - \frac{π}{2} C (2 \frac{2}{π} x) + C

\frac{8 s + 16}{s ^{2} + 8 s + 16}

x \to \infty lim (\frac{cos ( 2 x )}{x ^{2}})

\frac{d}{d x} (3 x - 5 cos^{2} (π x))

4 tan (x^{3} + 2 x^{2})

\int \frac{1}{x ^{2} + 6 x + 34} d x

What is the integral of (1+sin^2(x))/(sqrt(x)) ?
The integral of (1+sin^2(x))/(sqrt(x)) is 3sqrt(x)-(sqrt(pi))/2 \C(sqrt(2)sqrt(2/pi)sqrt(x))+C