What is the integral of x^5sin(x^2) ?

The integral of x^5sin(x^2) is -1/2 x^4cos(x^2)+x^2sin(x^2)+cos(x^2)+C

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integral of x^5sin(x^2)

\int x^{5} sin (x^{2}) d x

- \frac{1}{2} x^{4} cos (x^{2}) + x^{2} sin (x^{2}) + cos (x^{2}) + C

Solution steps

\int x^{5} sin (x^{2}) d x

Apply u-substitution

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

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

Apply Integration By Parts

: - u^{2} cos (u) - \int - 2 u cos (u) d u

= \frac{1}{2} (- u^{2} cos (u) - \int - 2 u cos (u) d u)

\int - 2 u cos (u) d u = - 2 (u sin (u) + cos (u))

= \frac{1}{2} (- u^{2} cos (u) - (- 2 (u sin (u) + cos (u))))

Substitute back

u = x^{2}

= \frac{1}{2} (- (x^{2})^{2} cos (x^{2}) - (- 2 (x^{2} sin (x^{2}) + cos (x^{2}))))

Simplify

\frac{1}{2} (- (x^{2})^{2} cos (x^{2}) - (- 2 (x^{2} sin (x^{2}) + cos (x^{2})))) : - \frac{1}{2} x^{4} cos (x^{2}) + x^{2} sin (x^{2}) + cos (x^{2})

= - \frac{1}{2} x^{4} cos (x^{2}) + x^{2} sin (x^{2}) + cos (x^{2})

Add a constant to the solution

= - \frac{1}{2} x^{4} cos (x^{2}) + x^{2} sin (x^{2}) + cos (x^{2}) + C

y = \frac{x + 1}{5 - x}, (7, - 4)

\frac{d}{d x} (x^{2} cos (\frac{1}{x ^{2}}))

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

y = x, y = 0, x = 25

y^{^{'}} + 2 y = 3 x

What is the integral of x^5sin(x^2) ?
The integral of x^5sin(x^2) is -1/2 x^4cos(x^2)+x^2sin(x^2)+cos(x^2)+C