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

The integral of (xcos(x^2))/(sqrt(sin(x^2))) is sqrt(sin(x^2))+C

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integral of (xcos(x^2))/(sqrt(sin(x^2)))

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

sin (x^{2}) + C

Solution steps

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

Apply u-substitution

: \int \frac{cos ( u )}{2 sin ( u )} d u

= \int \frac{cos ( u )}{2 sin ( u )} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{cos ( u )}{sin ( u )} d u

Apply u-substitution

: \int \frac{1}{v} d v

= \frac{1}{2} \cdot \int \frac{1}{v} d v

Apply the Power Rule

: 2 v^{\frac{1}{2}}

= \frac{1}{2} \cdot 2 v^{\frac{1}{2}}

Substitute back

= \frac{1}{2} \cdot 2 sin^{\frac{1}{2}} (x^{2})

Simplify

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

= sin (x^{2})

Add a constant to the solution

= sin (x^{2}) + C

y^{^{''}} + 26 y^{^{'}} + 6 y = 0

\int_{0}^{1} (e^{2 x} - e^{x}) d x

y^{^{''}} + 64 y^{^{'}} = 0

\int_{1}^{3} (x^{2} + 2 x) d x

y^{^{'}} + x^{2} sec (y) = 0

What is the integral of (xcos(x^2))/(sqrt(sin(x^2))) ?
The integral of (xcos(x^2))/(sqrt(sin(x^2))) is sqrt(sin(x^2))+C