Step 1: First, we take the derivative of the outside function and ignore the inside functions. The
derivative of u^3 is 3u^2.
We get [u]^3 = 3[u]^2.
Step 2: Next, take the derivative of the cosine term and multiply. The derivative of cos u is −
sin u.
[cos(u)]^3 = −3[cos(u)]^2 sin(u)
Step 3: Finally, we take the derivative of x + 1 and multiply. The derivative of x + 1 is 1.
[cos(x + 1)]^3 = −3[cos(x + 1)]^2 sin(x + 1)
- B We can do this integral with u-substitution.
Step 1: Let u = x + 3. Then du = dx and u − 3 = x.
Step 2: Substituting, we get
Why is this better than the original integral, you might ask? Because now we can distribute and
the integral becomes easy.
Step 3: When we distribute, we get
Step 4: Now we can integrate.
Step 5: Substituting back, we get
