Calculus • Step-by-Step

Integral of sin(x) dx with Steps

Find the integral of sin(x) step by step. Learn why the antiderivative of sine is -cos(x) + C with clear trigonometric integration rules and steps.

Problem Statement
∫sin⁡(x) dx\int \sin(x) \, dx
Final Result
F(x)=−cos(x)+CF(x) = -\mathrm{cos}\left(x\right) + C

Formula Required: Sine Integration Rule

Because d/dx[cos(x)] = -sin(x), the antiderivative of sin(x) must be -cos(x).

∫sin⁡(x) dx=−cos⁡(x)+C\int \sin(x) \, dx = -\cos(x) + C

Step-by-Step Solution

2 Steps
1

Set up the indefinite integral with respect to variable $x$.

∫(sin(x)) dx\int \left( \mathrm{sin}\left(x\right) \right) \, dx
2

Apply the Sine Integration to find the antiderivative.

F(x)=−cos(x)F(x) = -\mathrm{cos}\left(x\right)

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Common Questions About Integral of sin(x) dx

Why is the integral of sin(x) negative cos(x)?

Since the derivative of cos(x) is -sin(x), reversing differentiation requires a negative sign: d/dx[-cos(x)] = sin(x).

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