What is the integral of (ye^{2xy}+x) ?

The integral of (ye^{2xy}+x) is (x^2)/2+1/2 e^{2yx}+C

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integral of (ye^{2xy}+x)

\int (y e^{2 x y} + x) d x

\frac{x ^{2}}{2} + \frac{1}{2} e^{2 y x} + C

Solution steps

\int y e^{2 x y} + x 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 x d x + \int y e^{2 y x} d x

\int x d x = \frac{x ^{2}}{2}

\int y e^{2 y x} d x = \frac{1}{2} e^{2 y x}

= \frac{x ^{2}}{2} + \frac{1}{2} e^{2 y x}

Add a constant to the solution

= \frac{x ^{2}}{2} + \frac{1}{2} e^{2 y x} + C

\frac{d y}{d x} = (y + 3)

y = x^{3} - 5 x + 1, (3, 13)

f^{^{''}} (x) = \frac{4}{x ^{2}}

\int \frac{1}{x ^{2} 49 x ^{2} + 16} d x

\frac{d}{d t} (e^{t a n (π t)})

What is the integral of (ye^{2xy}+x) ?
The integral of (ye^{2xy}+x) is (x^2)/2+1/2 e^{2yx}+C