Integration

\( { \rm If} \ u = 2v^{-3} +6v, { \rm \ integrating} \ u \ {\rm with \ respect \ to} \ v\ {\rm gives}\)

• \( -v^{-2} +3v^2 +C, \ {\rm where} \ C \ {\rm is \ an \ unknown \ constant.} \)

• \( -0.5v^{-4} +C, \ {\rm where} \ C \ {\rm is \ an \ unknown \ constant.} \)

• \( -0.5v^{-4} +12v^2\)

• \( -4v^{-2} +12v^2 +C, \ {\rm where} \ C \ {\rm is \ an \ unknown \ constant.} \)

• \( -v^{-4} +C, \ {\rm where} \ C \ {\rm is \ an \ unknown \ constant.} \)

\( {\rm Calculate} \ J \ {\rm where} \ J= \int^{0.90}_{0}(17000 +12000x - 28000x^{2})dx \)

• \(1.3 \times 10^{4} \)

• \( -1.9\times 10^{3} \)

• \( 4.6\times 10^{3} \)

• \( 2.7\times 10^{4} \)

• \( 1.2\times 10^{4} \)

\( { \rm If} \ y = 3e^{4x}, { \rm \ integrating} \ y \ {\rm with \ respect \ to} \ x\ {\rm gives}\)

• \( \frac{3}{4}e^{4x} +C, \ {\rm where} \ C \ {\rm is \ an \ unknown \ constant.} \)

• \( 12e^{4x} +C, \ {\rm where} \ C \ {\rm is \ an \ unknown \ constant.} \)

• \( \frac{3}{4}e^{4x+C}, \ {\rm where} \ C \ {\rm is \ an \ unknown \ constant.} \)

• \( \frac{3}{5}e^{5x} +C, \ {\rm where} \ C \ {\rm is \ an \ unknown \ constant.} \)

• \( \frac{3}{4}e^{4x+1} +C, \ {\rm where} \ C \ {\rm is \ an \ unknown \ constant.} \)

\( { \rm If} \ y = 2{\rm cos}(4x+\pi), { \rm \ integrating} \ y \ {\rm with \ respect \ to} \ x\ {\rm gives}\)

• \( \frac{1}{2}{\rm sin}(4x+\pi) +C, \ {\rm where} \ C \ {\rm is \ an \ unknown \ constant.} \)

• \( 2{\rm sin}(4x+\pi) +C, \ {\rm where} \ C \ {\rm is \ an \ unknown \ constant.} \)

• \( -\frac{1}{2}{\rm sin}(4x+\pi) +C, \ {\rm where} \ C \ {\rm is \ an \ unknown \ constant.} \)

• \( -8{\rm sin}(4x+\pi) +C, \ {\rm where} \ C \ {\rm is \ an \ unknown \ constant.} \)

• \( \frac{1}{2}{\rm cos}(4x+\pi) +C, \ {\rm where} \ C \ {\rm is \ an \ unknown \ constant.} \)

\( { \rm The \ integral \ } \int^{3}_{0}(x(3-x) dx { \rm \ gives \ the \ area \ of \ which \ area \ highlighted \ below?}\)