2. Questions Bank
  3. JEE Main & Advanced
  4. Mathematics
  5. Indefinite Integrals

JEE Main & Advanced Mathematics Indefinite Integrals Question Bank Critical Thinking

  • question_answer
    If \[u=\int_{{}}^{{}}{{{e}^{ax}}\cos bx\ dx}\] and \[v=\int_{{}}^{{}}{{{e}^{ax}}\sin bx\ dx}\], then \[({{a}^{2}}+{{b}^{2}})({{u}^{2}}+{{v}^{2}})=\]

    A) \[2{{e}^{ax}}\]

    B) \[({{a}^{2}}+{{b}^{2}}){{e}^{2ax}}\]

    C) \[{{e}^{2ax}}\]

    D) \[({{a}^{2}}-{{b}^{2}}){{e}^{2ax}}\]

    Correct Answer: C

    Solution :

    • \[u=\int_{{}}^{{}}{{{e}^{ax}}\cos bx\,dx}\]\[={{e}^{ax}}\frac{\sin bx}{b}-\frac{a}{b}\int_{{}}^{{}}{{{e}^{ax}}.\sin bx\,dx}\]                   
    • \[=\frac{{{e}^{ax}}\sin bx}{b}-\frac{a}{b}v\] \[\Rightarrow bu+av={{e}^{ax}}\sin bx\]  ..?(i)                   
    • Similarly \[bv-au=-{{e}^{ax}}\cos bx\]                 ..?(ii)                   
    • Squaring (i) and (ii) and adding, we get                
    • \[({{a}^{2}}+{{b}^{2}})({{u}^{2}}+{{v}^{2}})={{e}^{2ax}}\].


You need to login to perform this action.
You will be redirected in 3 sec spinner