Step-by-Step Solution | Integration of sin(x) using Substitution

integral of sinx

The integral of sin(x) can be found using the technique of integration by substitution

The integral of sin(x) can be found using the technique of integration by substitution. Here’s a step-by-step solution:

Step 1: Start with the integral expression ∫sin(x) dx.

Step 2: Identify a new variable u and set it equal to an expression that can be easily differentiated. In this case, let’s choose u = cos(x), since the derivative of cos(x) with respect to x is -sin(x), which matches the original function.

So, differentiate both sides of the equation u = cos(x) with respect to x to find du/dx = -sin(x).

Step 3: Solve the equation from Step 2 for dx in terms of du:
dx = du / (-sin(x)).

Step 4: Substitute the expressions for u and dx in the integral from Step 1:
∫sin(x) dx = ∫sin(x) (-sin(x)) du = -∫sin^2(x) du.

Step 5: Simplify -∫sin^2(x) du:
Using the trigonometric identity sin^2(x) = (1 – cos(2x)) / 2, we can rewrite the integral expression:
-∫sin^2(x) du = -∫(1 – cos(2x)) / 2 du.

Step 6: Distribute the -1/2 outside the integral and split the integral into two separate parts:
-1/2 ∫(1 – cos(2x)) du = -1/2 ∫du + 1/2 ∫cos(2x) du.

Step 7: Integrate each part separately:
-1/2 ∫du = -1/2 u + C1,
where C1 is the constant of integration.

1/2 ∫cos(2x) du = 1/2 ∫cos(2x) d(2x).
Using the chain rule, we have d(2x) = 2 dx, so this simplifies to:
1/2 ∫cos(2x) d(2x) = 1/2 ∫cos(2x) (2 dx) = ∫cos(2x) dx.

Step 8: Substitute the original variable x back in:
-1/2 ∫du + 1/2 ∫cos(2x) du = -1/2 u + C1 + 1/2 ∫cos(2x) dx.

Step 9: Recall that u = cos(x). Substitute back in for u:
-1/2 u + C1 + 1/2 ∫cos(2x) dx = -1/2 cos(x) + C1 + 1/2 ∫cos(2x) dx.

Step 10: Solve the integral ∫cos(2x) dx:
To integrate cos(2x), we can use the substitution v = 2x. This gives dv/dx = 2, so dx = dv/2.
Substituting this in, we have:
-1/2 cos(x) + C1 + 1/2 ∫cos(2x) dx = -1/2 cos(x) + C1 + 1/2 ∫cos(v) (dv/2) = -1/2 cos(x) + C1 + 1/4 ∫cos(v) dv.

Step 11: Integrate ∫cos(v) dv:
The integral of cos(v) is sin(v), so:
-1/2 cos(x) + C1 + 1/4 ∫cos(v) dv = -1/2 cos(x) + C1 + 1/4 sin(v) + C2,
where C2 is another constant of integration.

Step 12: Substitute back in for v = 2x and simplify:
-1/2 cos(x) + C1 + 1/4 sin(v) + C2 = -1/2 cos(x) + C1 + 1/4 sin(2x) + C2.

So, the final solution to the integral of sin(x) is:
∫sin(x) dx = -1/2 cos(x) + C1 + 1/4 sin(2x) + C2, where C1 and C2 are constants of integration.

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