hi
im totally blanking
how do i find the antiderivative of cos(0.5(x^2))?
thanks!
Hi! Don't worry, I can help you find the antiderivative of cos(0.5(x^2)). To do this, we can use a technique called integration by substitution. Here's how you can proceed:
Step 1: Identify the inside function
In this case, the inside function is 0.5(x^2). Let's label it as u.
u = 0.5(x^2)
Step 2: Calculate the derivative of u
To find the derivative of u, we'll need to use the chain rule. Differentiating u with respect to x gives:
du/dx = d(0.5(x^2))/dx
= 0.5 * 2x
= x
Step 3: Solve for dx
Rearrange the equation that we obtained in Step 2 to solve for dx:
dx = du/x
Step 4: Rewrite the integral in terms of u and dx
Substitute the expression for dx from Step 3 into the original integral:
∫cos(0.5(x^2)) dx = ∫cos(u) * (du/x)
Step 5: Simplify the integral
Notice that we can cancel out the x terms:
∫cos(u) * (du/x) = ∫cos(u) du
Now, we have a simpler integral to solve.
Step 6: Evaluate the integral
The antiderivative of cos(u) is sin(u):
∫cos(u) du = sin(u) + C
Where C is the constant of integration.
Step 7: Convert back to the original variable
Now, we need to convert the answer from Step 6 back into the original variable x:
sin(u) = sin(0.5(x^2))
So, the antiderivative of cos(0.5(x^2)) is sin(0.5(x^2)) + C, where C is the constant of integration.
I hope this explanation helps you understand how to find the antiderivative of cos(0.5(x^2))! Let me know if you have any other questions.