Home
Class 12
MATHS
Evaluate the following integral: int0^(p...

Evaluate the following integral: `int_0^(pi//2)x^2sinx\ dx`

Text Solution

Verified by Experts

Given,I=`int_0^(pi//2)x^2sinx\ dx`
=`x^2int_0^(pi//2)sinxdx+int_0^(pi//2)2xcosx dx`
=`x^2[−cosx]_0^(​π/2)​+(2xsinx)_0​^(π/2)​−int_0^(pi//2)2sinxdx`
=`−π/4​[cos(pi/2)​−cos0]+(2(pi/2)​sin(π/2)​−0)+(2cosx)_0^(π/2)`
​ =`−π/4​[0−1]+π+2(cos(π/2)​−cos0)`
=`π^2/4+π−2`
Promotional Banner

Topper's Solved these Questions

  • CONTINUITY

    RD SHARMA|Exercise Solved Examples And Exercises|277 Videos

  • DERIVATIVES AS A RATE MEASURER

    RD SHARMA|Exercise Solved Examples And Exercises|149 Videos

Similar Questions

Explore conceptually related problems

Evaluate the following integral: int_0^(pi//2)x^2cos2x\ dx

Evaluate the following integral: int_0^(pi//2)2s in x\ cos xtan^(-1)(sinx)dx

Evaluate the following integral: int_0^(pi//2)x\ s in xcos^2x\ dx

Evaluate the following integral: int_0^(pi//2)|sin x-cos x|dx

Evaluate the following integral: int_0^(pi//2)(xsinxcosx)/(sin^4x+cos^4x)dx

Evaluate the following integral: int_0^(pi//2)(s in x cos x)/(cos^2x+3cos x+2)dx

Evaluate the following integral: int_0^(pi//2)1/(a^2sin^2x+b^2cos^2x)dx

Evaluate the following integral: int_0^(15)[x^2]dx

Evaluate the following integral: int_0^(pi//2)(x+sinx)/(1+cosx)dx

Evaluate the following integrals int_(0)^(pi//2)cos x dx

RD SHARMA-DEFINITE INTEGRALS-Solved Examples And Exercises
  1. Evaluate the following integral: int0^1(tan^(-1)x)/(1+x^2)dx

    Text Solution

    |

  2. Evaluate the following integral: int0^1x t a n^(-1)x\ dx

    Text Solution

    |

  3. Evaluate the following integral: int0^(pi//2)x^2sinx\ dx

    Text Solution

    |

  4. Evaluate the following integral: int0^1(1-x^2)/((1+x^2)^2)dx

    Text Solution

    |

  5. Evaluate the following integral: int0^(pi//2)(cos^2x)/(1+3sin^2x)dx

    Text Solution

    |

  6. Evaluate the following integral: int0^pi5(5-4costheta)^(1//4)sinthetad...

    Text Solution

    |

  7. Evaluate the following integral: int4^9(sqrt(x))/((30-x^(3/2))^2)dx

    Text Solution

    |

  8. Evaluate the following integral: int0^pisin^3x(1+2cos)(1+cos x)^2dx

    Text Solution

    |

  9. Evaluate the following integral: int0^(pi//2)2s in x\ cos xtan^(-1)(si...

    Text Solution

    |

  10. Evaluate the following integral: int0^(pi//2)s in2xtan^(-1)(sinx)dx

    Text Solution

    |

  11. int0^a xsqrt((a^2-x^2)/(a^2+x^2))dx

    Text Solution

    |

  12. Evaluate the following integral: int0^(pi//2)(sin x cos x)/(cos^2x+3co...

    Text Solution

    |

  13. Evaluate the following integral: int0^(1//2)1/((1+x^2)sqrt(1-x^2))dx

    Text Solution

    |

  14. Evaluate the following integral: int0^1(cos^(-1)x)^2dx

    Text Solution

    |

  15. Evaluate the following integral: int(pi//3)^(pi//2)(sqrt(1+cosx))/((1-...

    Text Solution

    |

  16. Evaluate the following integral: int(-a)^asqrt((a+x)/(a-x))dx

    Text Solution

    |

  17. Evaluate the following integral: int(1//3)^1((x-x^3)^(1//3))/(x^4)dx

    Text Solution

    |

  18. Evaluate the following integral: int0^(pi//2)sqrt(cos x-cos^3x)"\ "(se...

    Text Solution

    |

  19. Evaluate the following integral: int0^9f(x)dx ,\ w h e r e\ f(x){sinx ...

    Text Solution

    |

  20. Evaluate the following integral: int2^4f(x)dx ,\ w h e r e\ f(x)={7x+3...

    Text Solution

    |