What is the integral of sinh(x^2) ?

The integral of sinh(x^2) is 1/2 (-(sqrt(pi))/2 \erf(x)+(sqrt(pi))/2 \erfi(x))+C

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integral of sinh(x^2)

\int sinh (x^{2}) d x

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

Solution steps

\int sinh (x^{2}) d x

Rewrite using trig identities

= \int \frac{- e ^{- x^{2}} + e ^{x^{2}}}{2} d x

Take the constant out:

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

= \frac{1}{2} \cdot \int - e^{- x^{2}} + e^{x^{2}} 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

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

\int e^{- x^{2}} d x = \frac{π}{2} erf (x)

\int e^{x^{2}} d x = \frac{π}{2} erfi (x)

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

Add a constant to the solution

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

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