How do you integrate #int sqrt(x^2-25) dx# using trigonometric substitution?
Let suppose:
the integral become after suppose:
After simplified it we will get:
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To integrate ∫√(x^2 - 25) dx using trigonometric substitution, we can use x = 5 sec(θ). Then dx = 5 sec(θ) tan(θ) dθ. Substituting these expressions into the integral and simplifying yields ∫(5 sec(θ))^2 tan(θ) dθ. Simplify further to obtain ∫25 sec^2(θ) tan(θ) dθ. Using the identity sec^2(θ) = tan(θ) + 1, the integral becomes ∫25(tan(θ) + 1) tan(θ) dθ. This simplifies to ∫(25tan^2(θ) + 25) dθ. Integrate term by term to get 25∫tan^2(θ) dθ + 25∫dθ. Using the trigonometric identity tan^2(θ) = sec^2(θ) - 1, the integral becomes 25∫(sec^2(θ) - 1) dθ + 25∫dθ. Integrate each term to get 25(tan(θ) - θ) + 25θ + C. Substitute back x = 5 sec(θ) to obtain the final result: 25(tan^(-1)(x/5) - x/5) + 25 tan^(-1)(x/5) + C.
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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