Lesson Worksheet: Integration by Parts Mathematics • Higher Education

In this worksheet, we will practice using integration by parts to find the integral of a product of functions.


Determine ο„Έ2π‘₯𝑒π‘₯οŠ¨ο—οŠ°οŠ¨d.

  • A2π‘₯𝑒+οŠ¨ο—οŠ°οŠ¨C
  • B2ο€Ό12π‘₯βˆ’π‘₯+1οˆπ‘’+οŠ¨ο—οŠ°οŠ¨C
  • C4ο€Ό12π‘₯βˆ’π‘₯+1οˆπ‘’+οŠ¨ο—οŠ°οŠ¨C
  • D4ο€Ήπ‘₯βˆ’π‘₯+1𝑒+οŠ¨ο—οŠ°οŠ¨C
  • E4ο€Ό12π‘₯βˆ’π‘₯βˆ’1οˆπ‘’+οŠ¨ο—οŠ°οŠ¨C


Determine ο„Έ9π‘₯+7𝑒π‘₯οŠ«ο—d.

  • Aβˆ’95ο€Όπ‘₯+4445οˆπ‘’+οŠ±οŠ«ο—C
  • B95ο€Όπ‘₯+4445οˆπ‘’+οŠ±οŠ«ο—C
  • Cβˆ’15(9π‘₯+7)𝑒+οŠ±οŠ«ο—C
  • Dβˆ’95ο€Όπ‘₯+2645οˆπ‘’+οŠ±οŠ«ο—C
  • Eβˆ’9ο€Όπ‘₯+4445οˆπ‘’+οŠ±οŠ«ο—C


Determine ο„Έ(3π‘₯βˆ’5)π‘₯lnd.

  • A13(3π‘₯βˆ’5)[(3π‘₯βˆ’5)+1]+lnC
  • B13(3π‘₯βˆ’5)(3π‘₯βˆ’5)βˆ’1+lnC
  • C13(3π‘₯βˆ’5)[(3π‘₯βˆ’5)βˆ’1]+lnC
  • D13(3π‘₯+5)(3π‘₯βˆ’5)βˆ’π‘₯+lnC


Determine ο„Έο€Ί28π‘₯Γ·3√π‘₯π‘₯lnd.

  • A43√π‘₯[8π‘₯+2]+lnC
  • B23√π‘₯[8π‘₯βˆ’2]+lnC
  • C43√π‘₯8π‘₯βˆ’2+lnC
  • D43√π‘₯[8π‘₯βˆ’2]+lnC


Suppose ο„Έ(βˆ’6π‘₯βˆ’7)9π‘₯π‘₯=π‘¦π‘§βˆ’ο„Έπ‘§π‘¦lndd. Which of the following is equal to 𝑦𝑧?

  • Aβˆ’π‘₯(3π‘₯+7)9π‘₯ln
  • Bβˆ’π‘₯2(3π‘₯+14)+C
  • C(βˆ’6π‘₯βˆ’7)9π‘₯ln
  • Dβˆ’π‘₯(3π‘₯+7)


A curve passes through ο€Ό0,715 and the tangent at its point (π‘₯,𝑦) has slope 8π‘₯√2π‘₯+1. What is the equation of the curve?

  • A𝑦=815(2π‘₯+1)(3π‘₯βˆ’1)+1
  • B𝑦=815(2π‘₯+1)(3π‘₯βˆ’1)βˆ’115
  • C𝑦=415(2π‘₯+1)(8π‘₯βˆ’1)+1115
  • D𝑦=815(2π‘₯+1)(3π‘₯βˆ’1)βˆ’1615


The slope of the tangent to the curve 𝑦=𝑓(π‘₯) at the point (π‘₯,𝑦) is given by 3π‘₯𝑒(2π‘₯+1)οŠ¨ο—οŠ¨. Determine 𝑓(π‘₯) if the point ο€Ή1,5π‘’ο…οŠ¨ lies on the curve.

  • A𝑓(π‘₯)=βˆ’3𝑒4(2π‘₯+1)+214π‘’οŠ¨ο—οŠ¨
  • B𝑓(π‘₯)=𝑒4(2π‘₯+1)+1112π‘’οŠ¨ο—οŠ¨
  • C𝑓(π‘₯)=3𝑒4(2π‘₯+1)+194π‘’οŠ¨ο—οŠ¨
  • D𝑓(π‘₯)=3π‘₯𝑒4(2π‘₯+1)+194π‘’οŠ¨ο—οŠ¨


Use integration by parts to evaluate ο„Έπ‘₯π‘₯π‘₯sind.

  • Aπ‘₯π‘₯βˆ’π‘₯+cossinC
  • Bπ‘₯(π‘₯βˆ’π‘₯)+sincosC
  • CsincosCπ‘₯+π‘₯π‘₯+
  • Dβˆ’π‘₯βˆ’π‘₯π‘₯+sincosC
  • EsincosCπ‘₯βˆ’π‘₯π‘₯+


By setting 𝑒=𝑒 and dcosd𝑣=π‘₯π‘₯, evaluate 𝑒π‘₯π‘₯cosd by integrating by parts.

  • A2𝑒(π‘₯+π‘₯)+sincosC
  • B12𝑒(π‘₯βˆ’π‘₯)+sincosC
  • C𝑒(π‘₯+π‘₯)+sincosC
  • D2𝑒(π‘₯βˆ’π‘₯)+sincosC
  • E12𝑒(π‘₯+π‘₯)+sincosC


Integrate ο„Έπ‘₯π‘₯lnd by parts using 𝑒=π‘₯ln and dd𝑣=π‘₯.

  • AlnCπ‘₯βˆ’π‘₯+
  • Bπ‘₯π‘₯+1+lnC
  • Cπ‘₯(π‘₯+1)+lnC
  • Dπ‘₯(π‘₯βˆ’1)+lnC
  • Eπ‘₯π‘₯βˆ’1+lnC

This lesson includes 15 additional questions and 180 additional question variations for subscribers.

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