sinx + C
∫cosxdx
The expression sinx + C represents the indefinite integral of the trigonometric function sine. The constant C is known as the constant of integration, which is added to the result since there could be several functions whose derivative is equal to sine.
To find the indefinite integral of sinx, we can use the integration by substitution method. Let u = sin(x), then du/dx = cos(x), and dx = du/cos(x). Substituting these values in the integral gives:
∫ sin(x)dx = ∫ u * (du/cos(x))
= ∫ (u/cos(x))du
Now, we can substitute back for u:
= ∫ (sin(x)/cos(x))cos(x) dx
= ∫ tan(x)cos(x) dx
This integral can be solved by using integration by parts. Let u = tan(x) and dv = cos(x)dx, then du/dx = sec^2(x) and v = sin(x). Substituting these values in the integral gives:
∫ tan(x)cos(x) dx = tan(x)sin(x) – ∫ sin(x)sec^2(x) dx
Using the identity sec^2(x) = 1 + tan^2(x), we can simplify the second term:
∫ sin(x)sec^2(x) dx = ∫ sin(x)(1+tan^2(x)) dx
= ∫ sin(x)dx + ∫ sin(x)tan^2(x) dx
= -cos(x) – ∫ (sin(x))(sec^2(x) – 1)dx
= -cos(x) – tan(x) + ∫ sin(x) dx
= -cos(x) – tan(x) – cos(x) + C
Therefore, the indefinite integral of sin(x) is given by:
∫ sin(x)dx = -cos(x) – tan(x) + C, where C is the constant of integration.
More Answers:
Mastering Quadratic Functions: Understanding the Quadratic Parent Function for Accurate GraphsUnderstanding the Linear Parent Function: A Fundamental Concept in Mathematics and its Real-World Applications
Understanding Exponential Growth: Examples and Strategies for Management.