Solving the Definite Integral ∫ sin(x) dx | A Step-by-Step Guide using Integration by Substitution

∫ sinx dx

To solve the definite integral ∫ sin(x) dx, which represents the area under the curve of the function y = sin(x) with respect to x, we can use several methods

To solve the definite integral ∫ sin(x) dx, which represents the area under the curve of the function y = sin(x) with respect to x, we can use several methods. In this case, we will use the method of integration by substitution.

First, let’s rewrite the integral using the substitution u = sin(x). To do that, we need to find the derivative of u with respect to x.

Differentiating u = sin(x) with respect to x gives du/dx = cos(x). Rearranging this equation, we get dx = du/cos(x).

Now, let’s substitute these values into the integral:

∫ sin(x) dx = ∫ u * (du/cos(x))

We notice that cos(x) appears in the denominator, which is not convenient. To solve this, we can use the Pythagorean identity for trigonometric functions: cos^2(x) = 1 – sin^2(x).

Simplifying this equation, we find cos(x) = sqrt(1 – sin^2(x)). But we already know that u = sin(x). Substituting these values, cos(x) = sqrt(1 – u^2).

Now we can substitute all the expressions into the integral:

∫ sin(x) dx = ∫ u * (du / sqrt(1 – u^2))

This is now a simpler integral to solve. We can factor out the constant 1 from the square root expression:

∫ sin(x) dx = ∫ (1/sqrt(1 – u^2)) * u du

This integral can be recognized as the integral of arcsin(u):

∫ sin(x) dx = arcsin(u) + C

Finally, substitute back for u:

∫ sin(x) dx = arcsin(sin(x)) + C

Therefore, the solution to the integral of sin(x) dx is arcsin(sin(x)) + C, where C represents the constant of integration.

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