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Note: The incorrect answers have been deleted and the correct answer left in place.
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\noindent
1.
$$\int 4x^2 + 3x + 1\ dx = $$
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\item{(B)}
$${4\over 3} x^3 + {3\over 2} x^2 + x + C$$
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2.
$$\int x e^x\ dx = $$
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\item{(C)}
$$xe^x - e^x + C$$
(Check with the product rule for derivatives.)
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3.  
$$\int x (x^2 + 1)^{3/2}\ dx = $$
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\item{(D)}
$${1\over 5}(x^2 + 1)^{5/2} + C$$
(Check with the chain rule for derivatives.)
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4.  
$$\int x\cos(3x^2)\ dx = $$
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\item{(C)}
$${1\over 6}\sin(3x^2) + C$$
(Check with the chain rule for derivatives.)
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5.  
$$\int \sin(x)\cos(x)\ dx$$
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\item{(A)} 
$${-1\over 2} \cos^2(x) + C$$
\item{(C)}
$${1\over 2} \sin^2(x) + C$$
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.
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6.  
$$\int {1\over 1 + x^2}\ dx$$
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\item{(B)}
$$\tan^{-1}(x) + C = \arctan(x) + C$$
\bye
