Note: The incorrect answers have been deleted and the correct answer left in place. 1.

∫ 4x^{2} + 3x + 1 dx =

(B)

\frac{4}{3}x^{3} + \frac{3}{2}x^{2} + x + C

2.

∫ xe^{x} dx =

(C)

xe^{x} − e^{x} + C

(Check with the product rule for derivatives.) 3.

∫ x(x^{2} + 1)^{3/2} dx =

(D)

\frac{1}{5}(x^{2} + 1)^{5/2} + C

(Check with the chain rule for derivatives.) 4.

∫ x cos(3x^{2}) dx =

(C)

\frac{1}{6} sin(3x^{2}) + C

(Check with the chain rule for derivatives.) 5.

∫ sin(x) cos(x) dx

(A)

\frac{−1}{2} cos^{2}(x) + C

\frac{1}{2} sin^{2}(x) + C

Both are correct by the chain rule for derivatives. Note also that these are equivalent by the trig indentity

sin^{2}(x) + cos^{2}(x) = 1.

Every function in the form (A) can also be written in the form (C), but with a different constant value.

6.

∫ \frac{1}{1 + x^{2}} dx

(B)

tan^{−1}(x) + C = arctan(x) + C