sinx dx
To find the integral of sin(x) dx, we can use integration by substitution
To find the integral of sin(x) dx, we can use integration by substitution.
1. Let’s start by choosing a substitution. Let u = cos(x), and then find du/dx.
du/dx = -sin(x)
Now we need to manipulate the original equation so that we have an expression with du/dx.
We know that sin^2(x) + cos^2(x) = 1. From this, we solve for sin^2(x).
sin^2(x) = 1 – cos^2(x)
And if we take the square root of both sides:
sin(x) = √(1 – cos^2(x))
We can rewrite this equation as:
sin(x) = (1 – cos^2(x))^(1/2)
2. Next, we substitute u = cos(x) and du/dx = -sin(x) into the integral expression.
Integral of sin(x) dx = -∫√(1 – cos^2(x)) du
3. Now we need to simplify the expression under the square root sign. Since we have u = cos(x), we can rewrite the expression as:
√(1 – cos^2(x)) = √(1 – u^2)
4. Substituting this back into the integral, we get:
Integral of sin(x) dx = -∫√(1 – u^2) du
5. To evaluate the integral, we recognize that this is the integral of a square root function, which can be solved using a trigonometric substitution. Let’s substitute u = sin(t), which means du/dt = cos(t). This gives us:
Integral of sin(x) dx = -∫√(1 – u^2) du = -∫√(1 – sin^2(t)) cos(t) dt
6. Simplifying further, we have:
Integral of sin(x) dx = -∫cos(t)^2 dt
7. Using the trigonometric identity cos^2(t) = (1 + cos(2t)) / 2, we can rewrite the integral as:
Integral of sin(x) dx = -∫(1 + cos(2t))/2 dt
8. Distributing the 1/2, we get:
Integral of sin(x) dx = -∫(1/2) dt – ∫(1/2) cos(2t) dt
9. Integrating each term separately, we have:
Integral of sin(x) dx = -t/2 – (1/4) sin(2t) + C
10. Since we originally substituted u = cos(x), we need to convert everything back in terms of x. Recall that u = cos(x), so we have:
t = sin^(-1)(u)
sin(2t) = 2sin(t)cos(t) = 2u√(1 – u^2)
11. Substituting these back into our result, we get the final answer:
Integral of sin(x) dx = -sin^(-1)(u)/2 – (1/4) (2u√(1 – u^2)) + C
Integral of sin(x) dx = -sin^(-1)(cos(x))/2 – (1/4) (2cos(x)√(1 – cos^2(x))) + C
Integral of sin(x) dx = -sin^(-1)(cos(x))/2 – (cos(x)√(1 – cos^2(x))) + C
Thus, the integral of sin(x) dx is -sin^(-1)(cos(x))/2 – (cos(x)√(1 – cos^2(x))) + C, where C is the constant of integration.
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