Integration By Parts

Chapter 7

Class XII

Exercise 7.6

Q.No. 1. x.sinx

Solution:

From question it is clear that

I = ഽx.sinx.dx

Put First function = x
Second function = sinx
Now, applying integration by parts;
I = - x.cosx + ഽcosx.dx
or, - x.cosx + sinx + C Ans.

Q.No. 2. x.sin3x

Solution:

From question it clear that

I = ഽx.sin3x.dx
Put first function = x
Second function = sin3x
Now applying integration by parts
I = - x.cos3x/3 + ഽcos3x.dx/3
or, -x.cos3x/3 + sin3x/9 + C Ans.

Q.No. 3. x².ex

Solution:

From question it is clear that
 I = ഽx².ex.dx

Put first function = ex
Second function =  x²

Now applying integration by parts
I =  x².ex - ഽ2x.ex.dx
Again applying integration by parts
or,x².ex - 2x.ex +  ഽ2ex.dx
or,x².ex - 2x.ex + 2ex + C
or, ex(x² - 2x + 2) + C Ans.

Q.No. 4. x.logx

Solution:

From question it is clear that
I = ഽx.logx.dx
Put first function = logx
Second function = x

Now, applying integration by parts

I = x².logx/2 - ഽx.dx/2
or, x².logx/2 - x²/4 + C Ans.

Q.No. 5. x.log2x

Solution:

From question it is clear that
I = ഽx.log2x.dx

Put first function = log2x
Second function = x

Now applying integration by parts

I = log2x.x²/2 - ഽx².2/4x.dx
or, log2x.x²/2 - x²/4 + C Ans.

Q.No. 6. x².logx

Solution:

From question it is clear that
I = ഽ x².logx.dx

Put first function = logx
Second function = x²

Now applying integration by parts

I = logx. x³/3 - ഽx²dx/3
or, logx.x³/3 - x³/9 + C Ans.

Q.No. 7. xsin⁻¹x

Solution:

From question it is clear that
I =  ഽxsin⁻¹x.dx

Put first function = sin⁻¹x
Second function = x

Now applying integration by parts

I = sin⁻¹x. x²/2 - ഽx².dx/2√1-x²

or,  sin⁻¹x. x²/2+ഽ(1-x²-1).dx/√1-x²
or,  sin⁻¹x. x²/2+ ഽ√1-x².dx/2 -  ഽdx/2√1-x²
or,  sin⁻¹x. x²/2+ √1-x²/4 + sin⁻¹x./4 - sin⁻¹x./2 + C
or, (2x² - 1)sin⁻¹x./4 + √1-x²/4 +C Ans.

Q.No. 8 x.tan⁻¹x

Solution:

From question it is clear that
I = ഽx.tan⁻¹x

Put first function = tan⁻¹x
Second function = x

Now applying integration by parts

I = x².tan⁻¹x/2 - ഽx².dx/2(1 + x²)
or, x².tan⁻¹x/2 -ഽ(1 + x² - 1).dx/2(1 + x²)
or, x².tan⁻¹x/2 -ഽdx/2 + ഽdx/2(1 + x²)
or, x².tan⁻¹x/2 - x/2 + .tan⁻¹x/2 + C Ans.

Rest questions will be solved latter.