What is the integral of cos(x^2+1) ?

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

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integral of cos(x^2+1)

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

cos (1) \frac{π}{2} C (\frac{2}{π} x) - sin (1) \frac{π}{2} § (\frac{2}{π} x) + C

Solution steps

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

Rewrite using trig identities

= \int cos (x^{2}) cos (1) - sin (x^{2}) sin (1) 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 cos (x^{2}) cos (1) d x - \int sin (x^{2}) sin (1) d x

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

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

= cos (1) \frac{π}{2} C (\frac{2}{π} x) - sin (1) \frac{π}{2} § (\frac{2}{π} x)

Add a constant to the solution

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

y = x, 5 y = x + 6

\frac{d y}{d x} = \frac{4 y}{3 x}

\int \frac{1}{x ^{2} + 6 x + 5} d x

(8 (4 x)^{4} + 5)^{3}

\frac{d y}{d x} + y cos (x) = 6 cos (x), y (0) = 8

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