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Is the integral of sin/cos same as tan?

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Final answer:

The integral of sin(x)/cos(x) results in tan(x) plus a constant of integration, demonstrated by substitution in integration, and is supported by the derivative of tan(x) being sec^2(x).

Step-by-step explanation:

The integral of sin(x)/cos(x) is indeed the function tan(x) plus a constant of integration. To show this, we can use a substitution method where we let u = cos(x), then du = -sin(x)dx. When we substitute u into the integral, we have ∫ -1/u du, which is the natural logarithm of the absolute value of u, or ln|cos(x)|. The result of this integration is ln|sec(x)|, which is equivalent to tan(x) plus a constant, because the derivative of tan(x) is sec^2(x), and the integral of sec^2(x) is tan(x).

It's important to note that the trigonometric identities provided in the reference information, such as sin(a ± β) = sin(a)cos(β) ± cos(a)sin(β) or tan(a ± β) = (tan(a) + tan(β)) / (1 - tan(a)tan(β)), show various relationships between trigonometric functions but do not directly solve the integral of sin(x)/cos(x).

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