Math Answers

4 Find the
∫tan3xsec3xdx

tan2x+1

cot2x+1

sec2x+1

sec2x

5 Find
∫sec3xtanxdx

sin2x+c

cos2x+c

sec3x3+c

cos3x2+c

6 Evaluate
∫x+1x2−3x+2dx

3ln(x+2)−2ln(x+1)+c

3ln(x−2)−2ln(x−1)+c

−3ln(x−2)−2ln(x−1)+c

3ln(x−2)+2ln(x+1)+c

1 Evaluate
∫e4xdx

14e4x+c

ex+c

3ex3+c

13ex+c

2 Evaluate
∫xe6xdx

x6e6x−1136e6x+c

x3e6x+116e6x+c

x6e6x+1136e6x+c

−x6e6x+1136e6x+c

3 Find
∫xcosax2dx
with respect to x

cos3x+c

sin2x+c

sec2x+1

12asinax2+c

Show that an orthogonal map of the plane is either a reflection, or a rotation.

Apply the Gram-Schmidt diagonalisation process to find an orthonormal basis for the
subspace of C^4 generated by the vectors
{(1,−i,0,1),(−1,0, i,0),(−i,0,1,−1)}

Let T : P1 → P2 be defined by
T(a+bx) = b+ax+(a−b)x^2.
Check that T is a linear transformation. Find the matrix of the transformation with
respect to the ordered bases B1 = {1,x} and B2 = {x^2,x^2+x,x^2+x+1}. Find the
kernel of T . Further check that the range of T is
{a+bx+cx^2 ∈ P2 | a+c = b}.

Find the orthogonal canonical reduction of the quadratic form
−x2 + y2 + z2− 6xy− 6xz+ 2yz . Also, find its principal axes, rank and signature of the
quadratic form.

Let (x1, x2, x3) and (y1, y2, y3) represent the coordinates with respect to the bases
B1 = {(1,0,0),(0,1,0),(0,0,1)}, B2 = {(1,0,0),(0,1,2),(0,2,1)}.
If Q(X) = 2x21 +2x1x2 −2x2x3 +x22 +x23, find the representation of Q in terms of (y1, y2, y3).

Determine if the following function is even, odd, or neither.

f(x) = -9x4 + 5x + 3