Important Integration Questions With Answers | Class 12 Math Notes Study Material Download Free PDF

Integration questions with answers are available here for students of Class 11 and Class 12. Integration is an important topic for 11th and 12th standard students as these concepts are further covered in higher studies. The problems provided here are as per the CBSE board and NCERT curriculum. Exercising these questions will help students to solve the hard questions also and obtain more marks in the exam. Learn Integration Rules here.

The representation of the integration of a function is ∫f(x) dx. The common integral formulas used to solve integration problems are given below in the table.

\(\begin{array}{l}\int 1 d x=x+C\\\end{array} \) \(\begin{array}{l}\int a d x=a x+C\\\end{array} \) \(\begin{array}{l}\int x^{n} d x=\frac{x^{n+1}}{n+1}+C ; n \neq-1\\\end{array} \) \(\begin{array}{l}\int \sin x d x=-\cos x+C\\\end{array} \) \(\begin{array}{l}\int \cos x d x=\sin x+C\\\end{array} \) \(\begin{array}{l}\int \sec ^{2} x d x=\tan x+C\\\end{array} \) \(\begin{array}{l}\int \csc ^{2} x d x=-\cot x+C\\\end{array} \) \(\begin{array}{l}\int \sec x(\tan x) d x=\sec x+C\\\end{array} \) \(\begin{array}{l}\int \csc x(\cot x) d x=-\csc x+C\\\end{array} \) \(\begin{array}{l}\int \frac{1}{x} d x=\ln |x|+C\\\end{array} \) \(\begin{array}{l}\int e^{x} d x=e^{x}+C\\\end{array} \) \(\begin{array}{l}\int a^{x} d x=\frac{a^{x}}{\ln a}+C ; a>0, a \neq 1\end{array} \)

Questions on Integration with Solutions

Here are some questions based on the integration concept with solutions.

1. Integrate 1/(1+x²) for limit [0,1].

Solution:

\(\begin{array}{l}I=\int_{0}^{1} \frac{1}{1+x^{2}} d x\end{array} \)

\(\begin{array}{l}=\left[\tan ^{-1} x\right]_{0}^{1}\end{array} \)

\(\begin{array}{l}=\left[\tan ^{-1} 1-\tan ^{-1} 0\right]\end{array} \)

\(\begin{array}{l}=\left[\frac{\pi}{4}-0\right]\end{array} \)

\(\begin{array}{l}=\frac{\pi}{4}\end{array} \)

\(\begin{array}{l}\int_{0}^{1} \frac{1}{1+x^{2}} d x=\frac{\pi}{4}\end{array} \)

2. Find the value of ∫2x cos (x² – 5).

Solution: Let, I = ∫2xcos(x² – 5).dx

Let x² – 5 = t …..(1)

2x.dx = dt

Substituting these values, we have

I = ∫cos(t).dt

= sin t + c …..(2)

Substituting the value of 1 in 2, we have

= sin (x² – 5) + C

3. What is the value of ∫ 8 x³ dx.

Solution:

∫ 8 x³ dx = 8 ∫ x³ dx

= 8 x⁴ / 4 + C

= 2 x⁴ + C

4. Find the value of ∫ Cos x + x dx.

Solution: ∫ Cos x + x dx = ∫ Cos x dx + ∫ x dx

= sin x + x²/2 + C

5. ∫(x^e+e^x+e^e) dx

Solution: I = ∫(x^e+e^x+e^e) dx

Let us split the above equation.

∫x^e dx + ∫e^x dx + ∫e^e dx

By the formula, we know;

∫xⁿ dx = xⁿ⁺¹/n+1

Therefore,

x^e+1/e+1 + e^x + e^e x + C

Practice Questions

1. Integrate ∫ e^-x dx for [0,∞].
2. Integrate ∫x/(x+1) dx for [0,1]
3. Find ∫(ax²+bx+c) dx
4. Find ∫(2x²+e^x) dx
5. Find ∫[(x³+3x+4)/√x] dx
6. Evaluate ∫[(1-x)√x] dx
7. Evaluate ∫sec x(sec x+tan x) dx
8. Find the integration of 2x/1+x²
9. Find the integration of sin x cox(sin x)
10. What is the value of ∫[sin (ax+b) cos(ax+b)] dx.