Integral of sin(x) and Solutions Using Trigonometric Substitution

Integral of: sin(x) dx

The integral of sin(x) with respect to x is denoted as ∫sin(x) dx

The integral of sin(x) with respect to x is denoted as ∫sin(x) dx. To solve this integral, we can use integration by substitution or integration by parts.

Using integration by substitution, let’s make the substitution u = sin(x). Then, we can find du/dx by taking the derivative of u with respect to x, which is du/dx = cos(x). Rearranging this equation, we get dx = du/cos(x).

Now, substitute these values into the integral:

∫sin(x) dx = ∫u (du / cos(x))

Since dx = du/cos(x), we can simplify the integral:

∫sin(x) dx = ∫u (du / cos(x)) = ∫u / cos(x) du

Next, we need to replace the remaining cos(x) in the integral with u. We know that sin^2(x) + cos^2(x) = 1, so by rearranging, we can get cos(x) = √(1 – sin^2(x)) = √(1 – u^2).

Now, substitute these values into the integral:

∫sin(x) dx = ∫u / cos(x) du = ∫u / √(1 – u^2) du

At this point, we can use a trigonometric substitution. Let’s substitute u = sin(θ), which implies du = cos(θ) dθ. Also, keep in mind that sin^2(θ) + cos^2(θ) = 1, which means cos^2(θ) = 1 – sin^2(θ) = 1 – u^2.

Substituting these values into the integral:

∫sin(x) dx = ∫u / √(1 – u^2) du = ∫(sin(θ)) / √(1 – sin^2(θ)) cos(θ) dθ

Simplifying further:

∫sin(x) dx = ∫sin(θ) / √(cos^2(θ)) cos(θ) dθ = ∫sin(θ) / cos(θ) dθ

Now, we can recognize that sin(θ) / cos(θ) is equal to the derivative of the natural logarithm function ln(|sec(θ)|). Therefore, the integral becomes:

∫sin(x) dx = ∫sin(θ) / cos(θ) dθ = ∫d(ln(|sec(θ)|)) = ln(|sec(θ)|) + C

Finally, let’s substitute back u = sin(θ) into the solution:

∫sin(x) dx = ln(|sec(θ)|) + C

Remember that θ and x are related by u = sin(θ), so we need to find θ in terms of x. Since sin(θ) = u, we can solve for θ using the inverse sine function:

θ = arcsin(u)

Substituting this back into the solution:

∫sin(x) dx = ln(|sec(θ)|) + C = ln(|sec(arcsin(u))|) + C

Replacing u with sin(x):

∫sin(x) dx = ln(|sec(arcsin(sin(x)))|) + C

Simplifying further:

∫sin(x) dx = ln(|sec(arcsin(sin(x)))|) + C

The final result of the integral of sin(x) dx is ln(|sec(arcsin(sin(x)))|) + C, where C represents the constant of integration.

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