What is the integral of cos(8x^2+pi) ?

The integral of cos(8x^2+pi) is -(sqrt(pi))/4 \C((2*2x)/(sqrt(pi)))+C

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integral of cos(8x^2+pi)

\int cos (8 x^{2} + π) d x

- \frac{π}{4} C (\frac{2 \cdot 2 x}{π}) + C

Solution steps

\int cos (8 x^{2} + π) d x

Rewrite using trig identities

= \int - cos (8 x^{2}) d x

Take the constant out:

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

= - \int cos (8 x^{2}) d x

Apply Integral Substitution

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

Use the nonelementary integral:

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

= - \frac{1}{2} \cdot \frac{π}{2} C (\frac{2 u}{π})

Substitute back

u = 2 x

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

Simplify

- \frac{1}{2} \cdot \frac{π}{2} C (\frac{2 \cdot 2 x}{π}) : - \frac{π}{4} C (\frac{2 \cdot 2 x}{π})

= - \frac{π}{4} C (\frac{2 \cdot 2 x}{π})

Add a constant to the solution

= - \frac{π}{4} C (\frac{2 \cdot 2 x}{π}) + C

What is the integral of cos(8x^2+pi) ?
The integral of cos(8x^2+pi) is -(sqrt(pi))/4 \C((2*2x)/(sqrt(pi)))+C