Question

Use the substitution y = 5tan(θ) to find the integral of: a) ∫5sec^2(θ)dθ b) ∫5cot(θ)dθ c) ∫5sin(θ)dθ d) ∫5cos(θ)dθ

Answer 1

The substitution y = 5tan(θ) can be used to simplify the first integral ∫5sec^2(θ)dθ to y = 5tan(θ). The other integrals cannot be simplified using this specific substitution.

Step-by-step explanation:

To solve these integrals using the substitution y = 5tan(θ), we will use key trigonometric identities and properties of integrals. Heres the step by step breakdown:

1. For ∫5sec^2(θ)dθ: Since sec^2(θ) = 1 + tan^2(θ), we can rewrite the integral as ∫5(1+tan^2(θ))dθ. Now apply the substitution y = 5tan(θ). Then dy = 5sec^2(θ)dθ. Therefore, the integral becomes ∫dy = y = 5tan(θ).
2. For ∫5cot(θ)dθ: Using the identity cot(θ) = cos(θ)/sin(θ), this integral can be rewritten as ∫5cos(θ)/sin(θ) dθ. Unfortunately, with the substitution y = 5tan(θ), we cannot simplify this integral any further.
3. For ∫5sin(θ)dθ: Using the identity sin(θ) = tan(θ)/sec(θ), this integral becomes ∫5tan(θ)/sec(θ) dθ. Again, with the substitution y = 5tan(θ), this integral cannot be simplified further.
4. For ∫5cos(θ)dθ: Using the identity cos(θ) = 1/sec(θ), this integral can be rewritten as ∫5/sec(θ) dθ. The given substitution does not simplify this integral directly.

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