Understanding Integration | Using Trigonometric Identity to Integrate Cosine

∫cosudu

To integrate the function ∫cos(u) du, we can use the trigonometric identity of integration

To integrate the function ∫cos(u) du, we can use the trigonometric identity of integration. According to this identity, the integral of cosine can be expressed as the sine function:

∫cos(u) du = sin(u) + C,

where C is the constant of integration.

To understand how we arrived at this result, let’s go through the derivation:

We begin by using the basic integration rule for a constant factor. Since the derivative of sin(u) with respect to u is cos(u), we can rewrite the integral as:

∫cos(u) du = ∫1 * cos(u) du.

Next, we perform a substitution, letting v = sin(u). This implies that dv = cos(u) du. Applying this substitution, the integral becomes:

∫1 * cos(u) du = ∫1 * dv.

Simplifying, we have:

∫1 * dv = v + C,

where v is the substitution variable and C is the constant of integration.

Finally, recalling the substitution v = sin(u), we obtain the solution:

∫cos(u) du = sin(u) + C.

That is the final result.

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