NCERT Solutions For Class 12 Maths Chapter 7 Integrals

Calculus plays a crucial role in developing logical reasoning, analytical thinking, and problem-solving skills, and Chapter 7: Integrals is one of the most important and high-weightage chapters in Class 12 Maths. This chapter introduces students to integration as the inverse process of differentiation and helps them understand how to evaluate indefinite and definite integrals using standard methods.

NCERT Solutions for Class 12 Maths Chapter 7 – Integrals are prepared strictly according to the CBSE syllabus and exam pattern. These solutions include important board questions asked between 2018 and 2025 and are explained in a clear, step-by-step manner using simple language and proper mathematical presentation. This helps students build strong conceptual clarity, practise effectively, and score well in board examinations.

NCERT Solutions For Class 12 Maths Chapter 7 Integrals

Integrals
Difficult
Q.
Integrate the functions:
​x2logxx^2logx​​
Integrals
Medium
Q.
 Evaluatetheintegrals∫02xx+2dx \mathrm{Evaluate}\quad \mathrm{the}\quad \mathrm{integrals}\quad {\int }_{0}^{2}\mathrm{x}\sqrt{\mathrm{x}+2}{\hspace{0.17em}\hspace{0.17em}}\mathrm{dx} 
Integrals
Easy
Q.
 Integratethefunctionscos3xelogsinx \mathrm{Integrate}{\hspace{0.33em}}\mathrm{the}{\hspace{0.33em}}\mathrm{functions}{\hspace{0.33em}}{\mathrm{cos}}^{3}\mathrm{x}{{\hspace{0.33em}}}_{\mathrm{e}}\mathrm{logsinx} 
Integrals
Medium
Q.
 Integratethefunctions1cos(x+a)cos(x+b) \mathrm{Integrate}\quad \mathrm{the}\quad \mathrm{functions}\quad \frac{1}{\mathrm{cos}(\mathrm{x}+\mathrm{a})\mathrm{cos}(\mathrm{x}+\mathrm{b})} 
Integrals
Medium
Q.
 Integratethefunctionssin8x−cos8x1−2sin2xcos2x \mathrm{Integrate}\quad \mathrm{the}\quad \mathrm{functions}\frac{{\mathrm{sin}}^{8}\mathrm{x}-{\mathrm{cos}}^{8}\mathrm{x}}{1-2{\mathrm{sin}}^{2}{\mathrm{xcos}}^{2}\mathrm{x}} 
Integrals
Medium
Q.
 ∫0π2(2logsinx−logsin2x)dx {\int }_{0}^{\frac{\mathrm{\pi }}{2}}{\hspace{0.17em}}(2{\hspace{0.17em}}\mathrm{log}{\hspace{0.17em}}\mathrm{sin}{\hspace{0.17em}}\mathrm{x}{\hspace{0.17em}}-{\hspace{0.17em}}\mathrm{log}{\hspace{0.17em}}\mathrm{sin}{\hspace{0.17em}}2\mathrm{x}){\hspace{0.17em}}\mathrm{dx} 
Integrals
Medium
Q.
 ∫0π2cos5xsin5x+cos5xdx {\int }_{0}^{\frac{\mathrm{\pi }}{2}}\frac{{\mathrm{cos}}^{5}\mathrm{x}}{{\mathrm{sin}}^{5}\mathrm{x}+{\hspace{0.17em}}{\mathrm{cos}}^{5}\mathrm{x}}{\hspace{0.17em}}\mathrm{dx} 
Integrals
Difficult
Q.
 Evaluatetheintegrals∫12(1x−12x2)e2xdx \mathrm{Evaluate}\quad \mathrm{the}\quad \mathrm{integrals}\quad {\int }_{1}^{2}{\hspace{0.17em}}(\frac{1}{\mathrm{x}}-\frac{1}{2{\mathrm{x}}^{2}}){\hspace{0.17em}}{\mathrm{e}}^{2\mathrm{x}}{\hspace{0.17em}}\mathrm{dx} 
Integrals
Easy
Q.
 Evaluatetheintegrals∫0π2sinx1+cos2xdx \mathrm{Evaluate}{\hspace{0.33em}}\mathrm{the}{\hspace{0.33em}}\mathrm{integrals}{\hspace{0.33em}}{\int }_{0}^{\frac{\mathrm{\pi }}{2}}\frac{\mathrm{sinx}}{1+{\mathrm{cos}}^{2}\mathrm{x}}{\hspace{0.33em}dx} 
Integrals
Difficult
Q.
 Evaluatetheintegrals∫01sin−1(2x1+x2)dx \mathrm{Evaluate}\quad \mathrm{the}\quad \mathrm{integrals}\quad {\int }_{0}^{1}{\mathrm{sin}}^{-1}\left(\frac{2\mathrm{x}}{1+{\mathrm{x}}^{2}}\right){\hspace{0.17em}}\mathrm{dx} 
Integrals
Medium
Q.
Integrate the functions : ​xsin⁡−1xx\sin^{-1}x​​
Integrals
Medium
Q.
 ∫026x+3x2+4dx {\int }_{0}^{2}\frac{6\mathrm{x}+3}{{\mathrm{x}}^{2}+4}{\hspace{0.17em}}\mathrm{dx} 
Integrals
Easy
Q.
 ∫0π4(2sec2x+x3+2)dx {\int }_{0}^{\frac{\mathrm{\pi }}{4}}(2{\mathrm{sec}}^{2}{\hspace{0.17em}}\mathrm{x}+{\mathrm{x}}^{3}+2){\hspace{0.17em}}\mathrm{dx} 
Integrals
Easy
Q.
 ∫π6π4cosecxdx {\int }_{\frac{\mathrm{\pi }}{6}}^{\frac{\mathrm{\pi }}{4}}\mathrm{cosec}{\hspace{0.17em}}\mathrm{x}{\hspace{0.17em}}\mathrm{dx} 
Integrals
Easy
Q.
 ∫12(4x3−5x2+6x+9)dx {\int }_{1}^{2}(4{\mathrm{x}}^{3}-5{\mathrm{x}}^{2}+6\mathrm{x}+9)\mathrm{dx} 
Integrals
Medium
Q.
 ∫1−4x−x2dx \int \sqrt{1-4\mathrm{x}-{\mathrm{x}}^{2}}\mathrm{dx} 
Integrals
Medium
Q.
 Integrate the functions(x−3)ex(x−1)3 \begin{array}{l}{Integrate the functions}\\ \frac{(\mathrm{x}-3){\mathrm{e}}^{\mathrm{x}}}{{(\mathrm{x}-1)}^{3}}\end{array} 
Integrals
Medium
Q.
Integrate the functions x(logx)2x (logx)^2​​
Integrals
Difficult
Q.
Integrate the functions:
​xcos⁡−1xx\cos^{-1}x​​
Integrals
Easy
Q.
 Integratethefunctionsf(ax+b)[f(ax+b)]n \mathrm{Integrate}{\hspace{0.33em}}\mathrm{the}{\hspace{0.33em}}\mathrm{functions}{\hspace{0.33em}}\mathrm{f}{}\left(\mathrm{ax}+\mathrm{b}\right){\hspace{0.33em}}{\left[\mathrm{f}\left(\mathrm{ax}+\mathrm{b}\right)\right]}^{\mathrm{n}} 

Question 1

Evaluate the integral:

∫0π/2 [2 log(sin x) − log(sin 2x)] dx

Solution

We know that sin 2x = 2 sin x cos x.

log(sin 2x) = log 2 + log(sin x) + log(cos x)

Substituting in the given integral and simplifying, the terms
log(sin x) and log(cos x) cancel out over the interval (0, π/2).

Hence, the value of the integral is:

−(π/2) log 2

Question 2

Evaluate the integral:

∫0π/2 [cos5x / (sin5x + cos5x)] dx

Solution

Let the given integral be I.
Replacing x by (π/2 − x), we get another expression for I.

Adding both expressions:

2I = ∫0π/2 dx = π/2

Therefore,

I = π/4

Question 3

Evaluate:

∫02 x √(x + 2) dx

Solution

Let u = x + 2. Then x = u − 2.

Converting the limits and integrating, we obtain:

(32 + 16√2) / 15

Question 4

Evaluate:

∫12 [(1/x − 1/2x²) e2x] dx

Solution

Observe that:

d/dx [ e2x / (2x) ] = e2x(1/x − 1/2x²)

Hence, the value of the integral is:

e⁴/4 − e²/2

Question 5

Evaluate:

∫0π/2 [sin x / (1 + cos²x)] dx

Solution

Let u = cos x. Then du = −sin x dx.

The integral reduces to:

∫01 du / (1 + u²)

Therefore, the value is:

π/4

Question 6

Evaluate:

∫02 (6x + 3)/(x² + 4) dx

Solution

Splitting the integral and integrating termwise, we get:

3 log 2 + 3π/8

Question 7

Evaluate:

∫01 sin−1(2x / (1 + x²)) dx

Solution

Put x = tan θ. Then the integrand simplifies to 2θ.

After integration and applying limits, we obtain:

π/2 − log 2

Question 8

Evaluate:

∫π/6π/4 cosec x dx

Solution

∫ cosec x dx = log |tan(x/2)| + C

Hence, the value of the integral is:

log[(√2 − 1)/(2 − √3)]

Question 9

Evaluate:

∫12 (4x³ − 5x² + 6x + 9) dx

Solution

Integrating and applying limits, we get:

64/3

Question 10

Evaluate:

∫0π/4 (2 sec²x + x³ + 2) dx

Solution

Integrating termwise and applying limits, we obtain:

2 + π/2 + π⁴/1024

FAQs: NCERT Solutions for Class 12 Maths Chapter 7 – Integrals

Q1. Why is Chapter 7 Integrals important for Class 12 Maths?

Integrals is a core chapter of calculus and carries high weightage in CBSE board exams. It is also essential for understanding advanced topics in mathematics.

Q2. How many marks are usually asked from Integrals in board exams?

On average, 8 to 10 marks worth of questions are asked from this chapter, often as long-answer questions.

Q3. Are NCERT Solutions enough to score well in this chapter?

Yes. NCERT textbook questions and NCERT Solutions are more than sufficient for CBSE board exams, as most questions are directly or indirectly based on NCERT.

Q4. What are the most important topics in the Integrals chapter?

The most important topics include:

  • Standard and indefinite integrals

  • Methods of integration

  • Definite integrals and their properties

Q5. What is the difference between definite and indefinite integrals?

  • Indefinite integrals give a general solution and include a constant (+ C).

  • Definite integrals give a specific numerical value based on given limits.

Q6. Is Chapter 7 Integrals useful for competitive exams?

Yes. Concepts of integration are widely used in JEE and other competitive exams, especially in questions related to area, calculus, and applications of mathematics.