how do you integrate (cos^2 x)^4
To integrate the function (cos^2 x)^4, we can use the concept of integration by substitution. The substitution method involves replacing a part of the function with a new variable to simplify the integration process.
Step 1: Identify the substitution
Let's set u = cos^2 x. By doing this, we create a new variable u that simplifies the expression.
Step 2: Find du/dx and dx/du
Next, calculate du/dx by differentiating u with respect to x. In this case, the derivative of u = cos^2 x with respect to x is du/dx = -2cosxsinx. To find dx/du, we can rearrange the equation du/dx = -2cosxsinx to get dx = du / (-2cosxsinx).
Step 3: Rewrite the integral in terms of the new variable
Now, we can rewrite the original integral using the new variable u and the relationships dx = du / (-2cosxsinx) and (cos^2 x) = u:
∫ (cos^2 x)^4 dx = ∫ u^4 dx = ∫ (u^4)(du / (-2cosxsinx))
Step 4: Simplify the integral
Using the relationship (-2cosxsinx) = -sin(2x) and rewriting the integral, we have:
∫ (u^4)(du / (-2cosxsinx)) = ∫ (u^4)(du / (-sin(2x)))
Step 5: Evaluate the integral
Now that we have simplified the integral, we can proceed to evaluate it. The integral of (u^4)(du / (-sin(2x))) can be calculated by integrating u^4 with respect to u and evaluating sin(2x):
∫ (u^4)(du / (-sin(2x))) = -(1/2) ∫ u^4 du = -(1/2) * (u^5 / 5) + C
Step 6: Substitute back the original variable
Finally, substitute back the original variable u = cos^2 x to get the final result:
-(1/2) * (u^5 / 5) + C = -(1/2) * (cos^2 x)^5 / 5 + C
Therefore, the integral of (cos^2 x)^4 is -(1/10) * (cos^2 x)^5 + C, where C is the constant of integration.