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Space probes like Voyager 1, 2, New Horizons, etc, traveled beyond those gas giants, how did they cope up with their extreme gravity?

How was the trajectory of these probes unhindered by the immense gravity of those giant gas bodies? How was the trajectory decided?

Is it because of the small size of the space probes?

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    $\begingroup$ A trajectory that is not influenced by the immense gravity of those giant gas bodies is impossible. Very careful planning and a lot of numerical simulations allow to fly a trajectory that is acclerated by a swing by maneuver. Bo it does not depend on the size of the space probes. Probes of gigantic size would modify the orbit of the gas giants itself, but they are impossible to build and launch. Earth is a small pebble when compared with those gas giants. A probe of 10 % the Earth's mass is still too small. $\endgroup$ – Uwe Jun 11 '18 at 11:58
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    $\begingroup$ Math; the part that makes rocket science actually difficult. What they didn't know was if the circuitry would survive the immense radiation, and if it'd make it through the asteroid belt unscathed. $\endgroup$ – Mazura Jun 11 '18 at 23:13
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The trajectory was not only "unhindered" - it was enhanced!

Knowing mass of the planet you can calculate very precisely how the trajectory of a probe flying by will be affected. You modify the trajectory on arrival in such a way, that the departure trajectory will be exactly as desired. And due to some rather unintuitive physics caveats, you can make it so that the speed of the probe (relative to the Sun) at departure can be much higher than on arrival. This is called "gravity assist" or "Slingshot maneuver" and allows for some quite huge fuel savings. Voyagers performed good few gravity assists on their mission, and they are leaving the solar system faster than any other probe.

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    $\begingroup$ Unintuitive? When a slow-flying tennis ball hits a running locomotive, it bumps with much more speed. (Analogy by Randall Munroe). $\endgroup$ – kubanczyk Jun 11 '18 at 13:28
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    $\begingroup$ @kubanczyk: That's a very simplified analogy, which tends to be confusing because bumping against the front of the car is quite dissimilar of the gravitational swing behind the planet. $\endgroup$ – SF. Jun 11 '18 at 13:56
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    $\begingroup$ @kukis: Absolutely yes. Throwing tennis balls here on Earth also affects the path, orbit and speed of the planet, and the effect is about the same size. Planets are big. You can certainly do the math to work out how much energy was "stolen" by doing the integral of force applied over distance; that's the work, which has units of energy. The work that Jupiter did on Voyager results in a very, very small retrograde acceleration of Jupiter, and hence a very, very small lowering of its orbit. $\endgroup$ – Eric Lippert Jun 11 '18 at 19:13
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    $\begingroup$ @kukis: Law of preservation of momentum. Mass of planet * change of speed of planet = mass of spacecraft * change of speed of spacecraft. Gravity assist benefit is capped at half of planet's orbital speed, Jupiter's 13 km/s so at best 6.5km/s (likely much less). (721.9 kg * 6500m/s) / 1.9e27kg = 2.47e-21 m/s. That's 0.78 angstrom / millennium, $\endgroup$ – SF. Jun 11 '18 at 19:18
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    $\begingroup$ @SF: planetary.org/blogs/guest-blogs/2013/…: "during the Voyager encounters with Jupiter in 1979, Jupiter slowed down by roughly 10 to the -24th power kilometers per second", which is 1e-21m/s, so your approximation is definitely in the right ballpark! $\endgroup$ – Eric Lippert Jun 11 '18 at 19:25
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They did not !

This is the trajectory of Voyager 1 at Jupiter.

enter image description here credits wikipedia

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    $\begingroup$ @qwerty if by unhindered you mean “followed the intended path” yes. If by unhindered you mean that trajectory was unchanged then you are wrong. See the image: the deflection is significant $\endgroup$ – Antzi Jun 11 '18 at 10:40
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    $\begingroup$ @Antzi To be fair, the deflection was only significant because we shot the probe at the planet. If we just launched the probe "between the planets" the deflection would be much less pronounced (but I bet it would still tug over time!) $\endgroup$ – corsiKa Jun 11 '18 at 15:34
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    $\begingroup$ Note that this hyperbolic orbit is from the perspective of Jupiter, and not from the perspective of the sun. From the perspective of Jupiter, as you can see, Voyager's orbit is perfectly symmetrical. From the perspective of the Sun though, Voyager is being accelerated at a huge rate in the prograde Jupiter orbit direction at the close approach, and so there is a large acceleration from the perspective of the Sun. $\endgroup$ – Eric Lippert Jun 11 '18 at 19:27
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    $\begingroup$ @EricLippert yes, but it still looks like broken ellipses. I should add the graph too $\endgroup$ – Antzi Jun 11 '18 at 23:40
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    $\begingroup$ @Antzi - They're broken hyperbola (better: hyperbolic segments), not broken ellipses. The graph is fine. $\endgroup$ – David Hammen Jun 12 '18 at 6:44

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