Lesson: Integration by Parts for Indefinite Integrals

In this lesson, we will learn how to use integration by parts or partial integration for indefinite integrals to find the integral of a product of functions.

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Worksheet: Integration by Parts for Indefinite Integrals • 17 Questions • 1 Video

Q1:

Determine ๏„ธ 2 ๐‘ฅ ๐‘’ ๐‘ฅ ๏Šจ ๏— ๏Šฐ ๏Šจ d .

Q2:

Determine ๏„ธ 9 ๐‘ฅ + 7 ๐‘’ ๐‘ฅ 5 ๐‘ฅ d .

Q3:

Determine ๏„ธ ( 3 ๐‘ฅ โˆ’ 5 ) ๐‘ฅ l n d .

Q4:

Determine ๏„ธ ๏€บ 2 8 ๐‘ฅ รท 3 โˆš ๐‘ฅ ๏† ๐‘ฅ l n d .

Q5:

Suppose ๏„ธ ( โˆ’ 6 ๐‘ฅ โˆ’ 7 ) 9 ๐‘ฅ ๐‘ฅ = ๐‘ฆ ๐‘ง โˆ’ ๏„ธ ๐‘ง ๐‘ฆ l n d d . Which of the following is equal to ๐‘ฆ ๐‘ง ?

Q6:

A curve passes through ๏€ผ 0 , 7 1 5 ๏ˆ and the tangent at its point ( ๐‘ฅ , ๐‘ฆ ) has slope 8 ๐‘ฅ โˆš 2 ๐‘ฅ + 1 . What is the equation of the curve?

Q7:

The slope of the tangent to the curve ๐‘ฆ = ๐‘“ ( ๐‘ฅ ) at the point ( ๐‘ฅ , ๐‘ฆ ) is given by 3 ๐‘ฅ ๐‘’ ( 2 ๐‘ฅ + 1 ) 2 ๐‘ฅ 2 . Determine ๐‘“ ( ๐‘ฅ ) if the point ๏€น 1 , 5 ๐‘’ ๏… 2 lies on the curve.

Q8:

Use integration by parts to evaluate ๏„ธ ๐‘ฅ ๐‘ฅ ๐‘ฅ s i n d .

Q9:

By setting ๐‘ข = ๐‘’ ๏— and d c o s d ๐‘ฃ = ๐‘ฅ ๐‘ฅ , evaluate ๏„ธ ๐‘’ ๐‘ฅ ๐‘ฅ ๏— c o s d by integrating by parts.

Q10:

Integrate ๏„ธ ๐‘ฅ ๐‘ฅ l n d by parts using ๐‘ข = ๐‘ฅ l n and d d ๐‘ฃ = ๐‘ฅ .

Q11:

Determine ๏„ธ ( 5 ๐‘ฅ โˆ’ 1 2 ) ๐‘ฅ ๐‘ฅ s i n d .

Q12:

Determine ๏„ธ ( 3 ๐‘ฅ + 4 ) ๐‘’ ๐‘ฅ 2 ๐‘ฅ d .

Q13:

Determine ๏„ธ 2 ๐‘’ ๐‘ฅ 3 ( ๐‘ฅ + 1 ) ๐‘ฅ ๐‘ฅ 2 d .

Q14:

Determine ๏„ธ 6 4 ๐‘ฅ ๐‘ฅ l n d 5 .

Q15:

Determine ๏„ธ ๐‘ฅ ( 5 ๐‘ฅ ) ๐‘ฅ l n d ๏Šจ .

Q16:

Determine ๏„ธ 9 3 ๐‘ฅ ๐‘ฅ ๐‘ฅ l n d 6 .

Q17:

The slope of the tangent to the curve ๐‘ฆ = ๐‘“ ( ๐‘ฅ ) at the point ( ๐‘ฅ , ๐‘ฆ ) is given by 7 ๐‘ฅ ๐‘’ ( 2 ๐‘ฅ + 1 ) 2 ๐‘ฅ 2 . Determine ๐‘“ ( ๐‘ฅ ) if the point ๏€น 1 , 8 ๐‘’ ๏… 2 lies on the curve.
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