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# Class 12 RD Sharma Solutions – Chapter 19 Indefinite Integrals – Exercise 19.1

• Difficulty Level : Medium
• Last Updated : 02 Feb, 2021

### (i) âˆ« x4 dx

Solution:

âˆ« x4 dx = x4+1/(4+1) + Constant

= x5/5 + C

### (ii) âˆ« x5/4 dx

Solution:

âˆ« x5/4 dx = x5/4 + 1/(5/4 +1) + Constant

= 4/9 x9/4 + C

### (iii) âˆ« 1/x5 dx

Solution:

âˆ« 1/x5 dx = âˆ« x-5 dx

= x-5+1/(-5+1) + Constant

= x-4/(-4)+ C

= -1/(4x4) + C

### (iv) âˆ« 1/x3/2 dx

Solution:

âˆ« x-3/2 dx = x-3/2 + 1/(-3/2 +1) + Constant

= x-1/2/(-1/2) + C

= -2/(âˆšx)+ C

### (v) âˆ« 3x dx

Solution:

âˆ« 3x dx = 3x/log3 + Constant

### (vi) âˆ« 1/x2/3 dx

Solution:

âˆ« 1/x2/3 dx = âˆ« x-2/3 dx

= x-2/3 + 1/(-2/3+1) + Constant

= x1/3/(1/3) + C

= 3x1/3 + C

### (vii) âˆ« 32log3 x dx

Solution:

âˆ« 32log3 x dx =

= âˆ« x2 dx

= x2+1/(2+1) + Constant

= x3/3 + C

### (i)

Solution:

We know, cos 2x = 2cos2 x – 1

=

= âˆ«cos x dx

= sin x + Constant

### (ii)

Solution:

We know, cos 2x = 1 – 2sin2 x

= âˆ« sin x dx

= -cos x + Constant

### Question 3. Evaluate

Solution:

dx

We know, e loge x = x

= âˆ« x2 dx

= x2+1/2+1 + Constant

= x3/3 + C

### Question 4. Evaluate:

Solution:

= âˆ« a-x b-x dx

= âˆ« (ab)-x dx

= (ab)-x/loge (ab)-1 + Constant

= -a-x b-x/loge (ab) + C

or

= -a-x b-x/ ln(ab) + C

### (i)

Solution:

We know, cos 2x = 1 – 2sin2 x

= âˆ« 1/sin2x dx = âˆ« cosec2x dx

= -cot x + Constant

### (ii)

Solution:

We know, cos 2x = 2cos2 x – 1

= âˆ« 1/cos2 x dx = âˆ« sec2 x dx

= tan x + Constant

### Question 6. Evaluate: âˆ« elogâˆšx /x  dx

Solution:

âˆ« eloge âˆšx /x dx = âˆ«âˆšx/x dx

= âˆ« x-1/2 dx = x-1/2 + 1/(-1/2 + 1) + Constant

= x1/2 /(1/2) + C

= 2âˆšx + C

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