Worksheet: Integrals of Vector-Valued Functions

In this worksheet, we will practice integrating vector-valued functions in 2D and 3D.

Q1:

Calculate the integral (𝑡+3)𝑡ijd.

  • A𝑡2+3𝑡+ijC
  • B𝑡23+ijC
  • C𝑡23𝑡+ijC
  • D𝑡2+3𝑡+ijC
  • E𝑡2+3+ijC

Q2:

Evaluate the integral 𝑡+𝑡+3𝑡𝑡ijkd.

  • Aijk+4+32
  • Bijk432
  • C2+16ijk
  • D2+4+16ijk
  • E2+4+32ijk

Q3:

Calculate the integral 5𝑡4𝑡4𝑡+8+5𝑡+2𝑡ijkd.

  • A𝑡𝑡𝑡+8𝑡+54𝑡+2+ijkC
  • B4𝑡+𝑡+𝑡+8+54𝑡+2+ijkC
  • C𝑡𝑡𝑡+8𝑡+54𝑡+2𝑡+ijkC
  • D𝑡+4𝑡5𝑡8𝑡+45𝑡+2𝑡+ijkC
  • E5𝑡4𝑡𝑡+8𝑡+54𝑡+2𝑡+ijkC

Q4:

Evaluate the integral 2𝑡+(𝑡+1)𝑡𝑡ikd.

  • A23+45ik
  • B45+23ik
  • C1325+25615ik
  • D32+8ik
  • E1245+25615ik

Q5:

Evaluate the integral 𝑡+1𝑡+1+𝑒𝑡ijkd.

  • A43+2+1+1𝑒ijkln
  • B54+2+11𝑒ijkln
  • C34+2+1+1𝑒ijkln
  • D43+2+11𝑒ijkln
  • E34+2+11𝑒ijkln

Q6:

Evaluate the integral 𝑡𝑡+𝑡2𝑡+2𝑡2𝑡𝑡sectancossincosdijk.

  • A21𝜋814+16ijk
  • B21+𝜋8+1416ijk
  • C21+𝜋814+16ijk
  • D21+𝜋8+14+16ijk
  • E21𝜋8+14+16ijk

Q7:

Evaluate the integral 1𝑡+1+1𝑡+1+𝑡𝑡+1𝑡ijkd.

  • Alnln2+𝜋4+122ijk
  • Blnln2+𝜋4+132ijk
  • Clnln2𝜋4+132ijk
  • Dlnln2𝜋4132ijk
  • Elnln2+𝜋4122ijk

Q8:

Calculate the integral 11+𝑡+1𝑡1+11+𝑡𝑡ijkd.

  • A𝑡+(𝑡)+(𝑡)+tancoshsinhijkC
  • B𝑡+𝑡+𝑡+tancoshsinhijkC
  • C𝑡+𝑡𝑡+tancoshsinhijkC
  • D𝑡+(𝑡)(𝑡)+tancoshsinhijkC
  • E𝑡𝑡+𝑡+tancoshsinhijkC

Q9:

Calculate the integral 4𝑡+3𝑡+4𝑡5+4𝑡5𝑡+3𝑡ijkd.

  • A𝑡+𝑡+43𝑡5𝑡+𝑡53𝑡+3𝑡+ijkC
  • B𝑡+𝑡+34𝑡5+𝑡53𝑡+3𝑡+ijkC
  • C𝑡+𝑡+43𝑡5𝑡+𝑡35𝑡+3𝑡+ijkC
  • D𝑡+𝑡+34𝑡5𝑡+𝑡53𝑡+3𝑡+ijkC
  • E𝑡+𝑡+43𝑡5𝑡+𝑡35𝑡+3𝑡+ijkC

Q10:

Calculate the integral ((2𝑡))(2𝑡)+11+𝑡𝑡cossindijk.

  • A12(2𝑡)+(2𝑡)+𝑡+sincostanijkC
  • B12(2𝑡)+(2𝑡)+(𝑡)+sincostanijkC
  • C12(2𝑡)(2𝑡)+𝑡+cossincosijkC
  • D12(2𝑡)+(2𝑡)+𝑡+sincossinijkC
  • E12(2𝑡)+(2𝑡)+𝑡+sincoscosijkC

Q11:

Evaluate the integral (4𝑡+3)+(𝑡+3𝑡)𝑡ijd.

  • A38+88ij
  • B14+10ij
  • C22+28ij
  • D11+10ij
  • E22+443ij

Q12:

Calculate the integral 3𝑡+5𝑡+4𝑡𝑡ijkd.

  • A6𝑡+20𝑡+20𝑡+ijkC
  • B32𝑡+54𝑡+45𝑡+ijkC
  • C3𝑡+5𝑡+4𝑡+ijkC
  • D9𝑡+25𝑡+24𝑡+ijkC
  • E𝑡+𝑡+23𝑡+ijkC

Q13:

Calculate the integral [(5𝑡)+(3𝑡)]𝑡sincosdij.

  • A5(5𝑡)3(3𝑡)+cossinijC
  • B15(5𝑡)+13(3𝑡)+cossinijC
  • C15(5𝑡)13(3𝑡)+cossinijC
  • D(5𝑡)+(3𝑡)+cossinijC
  • E5(5𝑡)+3(3𝑡)+cossinijC

Q14:

Calculate the integral 1𝑡+6+1(2𝑡+5)+5𝑡𝑡+5𝑡ijkd.

  • Alnln|𝑡+6|+16(2𝑡+5)+|𝑡+5|+ijkC
  • Blnln|𝑡+6|16(2𝑡+5)+|𝑡+5|+ijkC
  • Clnln|𝑡+6|+110(2𝑡+5)+|𝑡+5|+ijkC
  • Dlnlnln|𝑡+6|+|(2𝑡+5)|+5𝑡|𝑡+5|+ijkC
  • Elnln|𝑡+6|13(2𝑡+5)+|𝑡+5|+ijkC

Q15:

Evaluate the integral (3(3𝑡))+(4(2𝑡))𝑡cossecdij.

  • A3+43ij
  • Bij+23
  • Cij34
  • D9+83ij
  • E9+643ij

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