Solving the Integral of sin(u) using Integration by Substitution

∫ sinu du

The integral of sin(u) with respect to u can be evaluated using integration by substitution

The integral of sin(u) with respect to u can be evaluated using integration by substitution. Let’s go through the steps:

Step 1: Identify the substitution.
In this case, we can let u = cos(x) since the derivative of cos(x) is -sin(x), which appears in the integrand.

Step 2: Compute the differential of u.
Differentiating both sides of the substitution equation u = cos(x), we get du = -sin(x)dx.

Step 3: Rewrite the integral in terms of u and du.
The given integral becomes ∫ sin(u) du = ∫ sin(u)((-1/sin(x)) du by substituting du = -sin(x)dx from Step 2.
Simplifying, we have ∫ -du.

Step 4: Evaluate the integral.
The integral of -1 with respect to u is simply -u.
So, the final result is -u + C, where C is the constant of integration.

Rewriting the answer in terms of the original variable x, we have:
∫ sin(u) du = -cos(x) + C.

Therefore, the integral of sin(u) du is -cos(x) + C, where C is a constant.

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