How To Integrate Cos(X): Step-By-Step Guide With Trigonometric Substitution And Chain Rule

cosx dx

sinx + c

The integral of cos(x) dx is equal to sin(x) + C, where C is the constant of integration.

To understand why, we can use the following steps:
1. Use the trigonometric identity cos^2(x) + sin^2(x) = 1 to express cos(x) in terms of sin(x): cos(x) = sqrt(1 – sin^2(x)).
2. Substitute u = sin(x) into the integral so that the integral becomes: ∫sqrt(1 – u^2) du.
3. Use the substitution method with u = sin(x) to find that du/dx = cos(x), so dx = du/cos(x).
4. Substitute dx with du/cos(x) in the integral so that we get: ∫sqrt(1 – u^2)/cos(x) du.
5. Recall that the secant function is equal to 1/cos(x), so we can write the integral as: ∫sqrt(1 – u^2)sec(x) du.
6. Recognize that this is the form of the integral for the derivative of the inverse sine function: d/dx[sin^-1(u)] = sqrt(1 – u^2).
7. Substitute u with sin(x) to get: ∫d/dx[sin^-1(sin(x))]sec(x) dx.
8. Use the chain rule to evaluate the derivative of sin^-1(sin(x)): d/dx[sin^-1(sin(x))] = d/dx[x] = 1.
9. Substitute the result into the integral, leaving us with: ∫sec(x) dx.
10. Use the substitution method with u = sin(x) to find that ∫sec(x) dx = ln|sec(x) + tan(x)| + C.
11. Substitute back sin(x) for u to get the final result: ∫cos(x) dx = sin(x) + C.

Thus, the integral of cos(x) dx is sin(x) + C.

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