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Q20:
Evaluate the definite integrals in Exercises 1 to 20.
20. $\int_{0}^{1} (x e^x + \sin \frac{\pi x}{4}) dx$
Solution :
Step 1: Decomposition of the Integral
By the linearity property of definite integrals, we can split the given integral into two distinct parts [Per the Linearity Property of Integrals]:
$\int_{0}^{1} (x e^x + \sin \frac{\pi x}{4}) \, dx = \int_{0}^{1} x e^x \, dx + \int_{0}^{1} \sin \frac{\pi x}{4} \, dx$
Step 2: Integration of the First Term ($I_1 = \int_{0}^{1} x e^x \, dx$)
To evaluate this integral, we employ the Integration by Parts formula: $\int u \, dv = uv - \int v \, du$ [Per the Integration by Parts Theorem].
- Let $u = x \implies du = dx$
- Let $dv = e^x \, dx \implies v = e^x$
Applying the formula:
$I_1 = [x e^x]_{0}^{1} - \int_{0}^{1} e^x \, dx$
$I_1 = (1 \cdot e^1 - 0 \cdot e^0) - [e^x]_{0}^{1}$
$I_1 = e - (e^1 - e^0) = e - e + 1 = 1$
Step 3: Integration of the Second Term ($I_2 = \int_{0}^{1} \sin \frac{\pi x}{4} \, dx$)
We use the standard integral formula $\int \sin(ax) \, dx = -\frac{1}{a} \cos(ax) + C$ [Per the Integration of Trigonometric Functions].
$I_2 = \left[ -\frac{1}{\pi/4} \cos \frac{\pi x}{4} \right]_{0}^{1}$
$I_2 = -\frac{4}{\pi} \left[ \cos \frac{\pi(1)}{4} - \cos \frac{\pi(0)}{4} \right]$
$I_2 = -\frac{4}{\pi} \left( \cos \frac{\pi}{4} - \cos 0 \right)$
Since $\cos \frac{\pi}{4} = \frac{1}{\sqrt{2}}$ and $\cos 0 = 1$:
$I_2 = -\frac{4}{\pi} \left( \frac{1}{\sqrt{2}} - 1 \right) = \frac{4}{\pi} \left( 1 - \frac{1}{\sqrt{2}} \right)$
Step 4: Summation of the Components
Combining $I_1$ and $I_2$:
Result = $1 + \frac{4}{\pi} \left( 1 - \frac{1}{\sqrt{2}} \right)$
Simplifying the expression within the parenthesis:
Result = $1 + \frac{4}{\pi} - \frac{4}{\pi\sqrt{2}} = 1 + \frac{4}{\pi} - \frac{2\sqrt{2}}{\pi}$
Final Solution: $1 + \frac{4 - 2\sqrt{2}}{\pi}$
More Questions from Class 12 Mathematics Integrals EXERCISE 7.8
- Q1: Evaluate the definite integrals in Exercises 1 to 20. 1. $\int_{-1}^{1} (x+1) dx$
- Q2: Evaluate the definite integrals in Exercises 1 to 20. 2. $\int_{2}^{3} \frac{1}{x} dx$
- Q3: Evaluate the definite integrals in Exercises 1 to 20. 3. $\int_{1}^{2} (4x^3 - 5x^2 + 6x + 9) dx$
- Q4: Evaluate the definite integrals in Exercises 1 to 20. 4. $\int_{0}^{\frac{\pi}{4}} \sin 2x dx$
- Q5: Evaluate the definite integrals in Exercises 1 to 20. 5. $\int_{0}^{\frac{\pi}{2}} \cos 2x dx$
- Q6: Evaluate the definite integrals in Exercises 1 to 20. 6. $\int_{4}^{5} e^x dx$
- Q7: Evaluate the definite integrals in Exercises 1 to 20. 7. $\int_{0}^{\frac{\pi}{4}} \tan x dx$
- Q8: Evaluate the definite integrals in Exercises 1 to 20. 8. $\int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \csc x dx$
- Q9: Evaluate the definite integrals in Exercises 1 to 20. 9. $\int_{0}^{1} \frac{dx}{\sqrt{1-x^2}}$
- Q10: Evaluate the definite integrals in Exercises 1 to 20. 10. $\int_{0}^{1} \frac{dx}{1+x^2}$
- Q11: Evaluate the definite integrals in Exercises 1 to 20. 11. $\int_{2}^{3} \frac{dx}{x^2-1}$
- Q12: Evaluate the definite integrals in Exercises 1 to 20. 12. $\int_{0}^{\frac{\pi}{2}} \cos^2 x dx$
- Q13: Evaluate the definite integrals in Exercises 1 to 20. 13. $\int_{2}^{3} \frac{x dx}{x^2+1}$
- Q14: Evaluate the definite integrals in Exercises 1 to 20. 14. $\int_{0}^{1} \frac{2x+3}{5x^2+1} dx$
- Q15: Evaluate the definite integrals in Exercises 1 to 20. 15. $\int_{0}^{1} x e^{x^2} dx$
- Q16: Evaluate the definite integrals in Exercises 1 to 20. 16. $\int_{1}^{2} \frac{5x^2}{x^2+4x+3} dx$
- Q17: Evaluate the definite integrals in Exercises 1 to 20. 17. $\int_{0}^{\frac{\pi}{4}} (2\sec^2 x + x^3 + 2) dx$
- Q18: Evaluate the definite integrals in Exercises 1 to 20. 18. $\int_{0}^{\pi} (\sin^2 \frac{x}{2} - \cos^2 \frac{x}{2}) dx$
- Q19: Evaluate the definite integrals in Exercises 1 to 20. 19. $\int_{0}^{2} \frac{6x+3}{x^2+4} dx$
- Q21: Choose the correct answer in Exercises 21 and 22. 21. $\int_{1}^{\sqrt{3}} \frac{dx}{1+x^2}$ equals
- Q22: Choose the correct answer in Exercises 21 and 22. 22. $\int_{0}^{\frac{2}{3}} \frac{dx}{4+9x^2}$ equals
CBSE Solutions for Class 12 Mathematics Integrals
- Integrals Class 12 EXERCISE 7.1
- Integrals Class 12 EXERCISE 7.10
- Integrals Class 12 EXERCISE 7.2
- Integrals Class 12 EXERCISE 7.3
- Integrals Class 12 EXERCISE 7.4
- Integrals Class 12 EXERCISE 7.5
- Integrals Class 12 EXERCISE 7.6
- Integrals Class 12 EXERCISE 7.7
- Integrals Class 12 EXERCISE 7.9
- Integrals Class 12 Miscellaneous Exercise on Chapter 7
Chapters in CBSE - Class 12 Mathematics
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