Practice Questions on Integration by Parts

Integration by parts is a technique used to integrate the product of two functions.

Formula: \int u\,dv = uv - \int v\,du

Solved Example

Question 1. Evaluate ∫2x cos(x)dx 

Solution: 

Given that, ∫2x cos(x)dx

According to the rule of integration by parts, 

Let us consider,

u = 2x and dv/dx = cos(x) 

Then

du/dx = 2  and  v = ∫cos(x) =  sin(x)

Now, using the formula for integration by parts;

∫u(dv/dx)dx = uv - ∫v(du/dx)dx

We get,

∫2x cox(x)dx = 2x sin(x) - ∫sinx.2dx

∫2x cos(x)dx = 2x sin(x) + 2cos(x) + c

where c is a constant.

Question 2. Evaluate ∫3x ln|x| dx

Solution:

Given, ∫3xln|x| dx 

According to rule of integration by parts, 

Let us consider,

u = ln|x|  and dv/dx = 3x

Then

du/dx = 1/x  and  v = ∫3xdx = 3x²/2

Now, using the formula for integration by parts,

∫u(dv/dx)dx = uv - ∫v(du/dx)dx

∫3x ln|x|dx = ln|x|.3x²/2 - ∫3x²/2(1/x)dx

= 3x²/2(ln|x|) - ∫3x/2 dx

= 3x²/2(ln|x|) - 3x²/4 + C

where C is a constant.

Question 3. Evaluate ∫10x²

Solution: 

Given that, ∫10x²

According to rule of integration by parts, 

Let us consider,

u = 10x² and dv/dx = 1

Then

du/dx = 20x  and  v = ∫(1)dx = x 

Now, using the formula for integration by parts;

∫u(dv/dx)dx = uv - ∫v(du/dx)dx

∫10x² dx = 10x²x - ∫x(20x)dx

∫10x² dx = 10x³ - ∫20x²dx

∫10x² dx = 10x³ - 20/3x³ + C

∫10x² dx = 10x³/3 + C

where C is a constant.

Question 4. Evaluate ∫x² e^4x dx

Solution:

Given that, ∫x² e^4x dx

According to rule of integration by parts, 

Let us consider,

u = x² and dv/dx = e^4x

Then

du/dx = 2x  and  v = ∫e^4x dx = 1/4(e^4x)

Then, using the formula for integration by parts,

∫u(dv/dx)dx = uv - ∫v(du/dx)dx

∫x²e^4x dx = (x²)1/4(e^4x) - ∫1/4(e^4x)(2x)dx

= 1/4(x²)(e^4x) - ∫2/4(x)(e^4x)dx 

= 1/4(x²)(e^4x) - ∫1/2(x)(e^4x)dx 

The resulting integral is still a product of 

two functions i.e. 1/2(x) and e^4x. 

So, We use the formula for integration again.

This time we take

u = 1/2(x)  and  dv/dx = e^4x

So

du/dx = 1/2  and  v = ∫e^4xdx = 1/4(e^4x)

Now, on putting these values in above equation,

∫x²e^4x dx = 1/4(x²)(e^4x) - ∫2/4(x)(e^4x)dx

= 1/4(x²e^4x) - {2/4(x).1/4(e^4x) - ∫(1/4(e^4x).2/4 dx}

= 1/4(x²e^4x) - 1/8(xe^3x) + 1/8(e^4x) + C

where C is a constant.

Question 5. Evaluate ∫ 2e^x sin(x) dx

Solution:

Given that, ∫ 2e^xsin(x) dx

According to rule of integration by parts, 

Let us consider,

u = 2e^x  and  dv/dx = sin(x)

Then

du/dx = 2e^x  and v = ∫sin(x)dx = -cos(x)

Now, using the formula for integration by parts,

∫u(dv/dx)dx = uv - ∫v(du/dx)dx

∫ 2e^xsin(x) dx = (2e^x)(-cos(x)) - ∫(-cos(x))(2e^x)dx

= -(2e^x)cos(x) + ∫ 2e^x cos(x)dx

The resulting integral is yet a product of two

functions i.e. 2e^x and cos(x). 

So, we use the formula for integration again.

This time we take

u = 2e^x  and  dv/dx = cos(x)

So

du/dx = 2e^x  and  v = ∫cos(x)dx = sin(x)

Now, on putting these values in the above equation, we get

∫ 2e^xsin(x) dx = -cos(x)(2e^x) + ∫ 2e^x cos(x)dx

= -(2e^x)cos(x) + {2e^xsin(x) - ∫sin(x)(2e^x)dx}

= -2e^xcos(x) + 2e^xsin(x) - ∫2e^xsin(x)dx

Now, notice that the integral ∫e^xsin(x)dx in the result

is same as the integral in the given question.

So, let us assume this as I.

Therefore,

I = ∫ 2e^xsin(x) dx

So,

I = 2e^xsin(x) - 2e^xcos(x) - I

2I = 2e^xsin(x) - 2e^xcos(x)

2I = 2(e^xsin(x) - e^xcos(x))

I = (e^xsin(x) - e^xcos(x))

So,

∫ 2e^xsin(x) dx = e^xsin(x) - e^xcos(x) + C

where C is a constant.

Question 6. Evaluate ∫2tan^-1x dx 

Solution:

Given, ∫2tan^-1x dx 

According to the rule of integration by parts, 

Let us consider,

u = tan^-1x  and  dv/dx = 2

Then

du/dx = 1/(1 + x²)  and  v = ∫(2)dx = 2x 

Now, using the formula for integration by parts,

∫u(dv/dx)dx = uv - ∫v(du/dx)dx

∫2tan^-1x dx = tan^-1x(2x) - ∫2x(1/(1+x²))dx 

let, p = (1 + x²)

So, dp/dx = 2x 

⇒ x = 1/2

Now, substituting the above values, we get

∫tan^-1x dx = tan^-1x(2x) - ∫(1/p)dx

= 2xtan^-1x - ∫(1/p)dx

= 2xtan^-1x - lnp

= 2xtan^-1x - ln(1 + x²) 

= 2xtan^-1x - ln√(1 + x²) + C

where C is a constant.

Practice Question

Question 1: Evaluate: ∫x sinx dx

Question 2: Evaluate: ∫x² sinx dx

Question 3: Evaluate: ∫x³e^xdx

Question 4: Evaluate: ∫x³cosx dx

Question 5. Evaluate: ∫xe^xcosx dx