(^) ∫eu du
Evaluate the integral to get
eu + C
Now substitute back to get
e^3 x
2
- C
B Use u-substitution. Here, u = 3x^3 + 2 and du = 9x^2 dx. Then, ∫ x^2 sin (3x^3 + 2) dx = (^) ∫ sin x dx
= (−cos u) + C = + C. Replace u for the final solution: + C.
C First take the derivative of f(x): f′(x) = . In order for f(x) to be
differentiable for all real values, both pieces of f(x) must be equal at x = −1 and both pieces off′(x) must be equal at x = −1. Therefore, plug x = −1 into both f(x) and f′(x): f(−1) = and f′(−1) = . When the two parts of f(−1) are set equal toeach other, 10 = −5a − b and when the two parts of f′(−1) are set equal to each other, −4 = 13a+ 3b. When this system is solved, a = −13 and b = 55.- C Here we need to use the Product Rule, which is
If f (x) = uv, where u and v are both functions of x,then f′(x) = u + vHere we getf′(x) = x^2 (−2 sin 2x) + 2x(cos 2x)- A Notice how this limit takes the form of the definition of the derivative, which is