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Q15:
Evaluate the definite integrals in Exercises 1 to 20.
15. $\int_{0}^{1} x e^{x^2} dx$
Solution :
Step 1: Setting up the Integral and Identifying the Method
We are tasked with evaluating the definite integral:
$I = \int_{0}^{1} x e^{x^2} dx$
Given the integrand is a composition of functions where the derivative of the exponent ($x^2$) is proportional to the multiplier ($x$), we employ the Method of Substitution [Per the substitution rule for definite integrals].
Step 2: Performing Algebraic Substitution
Let $u = x^2$.
Differentiating both sides with respect to $x$:
$\frac{du}{dx} = 2x \implies du = 2x dx \implies x dx = \frac{1}{2} du$
Step 3: Transforming the Limits of Integration
We must adjust the limits of integration from the variable $x$ to the variable $u$ using the relation $u = x^2$:
- When $x = 0$, $u = (0)^2 = 0$.
- When $x = 1$, $u = (1)^2 = 1$.
[By the Fundamental Theorem of Calculus, the transformation of bounds is required when performing change of variables].
Step 4: Substitution and Evaluation
Substituting $u$ and $du$ into the original integral:
$I = \int_{0}^{1} e^u \left( \frac{1}{2} du \right)$
$I = \frac{1}{2} \int_{0}^{1} e^u du$
Evaluating the antiderivative [Per the integration rule $\int e^u du = e^u + C$]:
$I = \frac{1}{2} [e^u]_{0}^{1}$
$I = \frac{1}{2} (e^1 - e^0)$
Step 5: Final Simplification
Since $e^0 = 1$ [By the zero exponent rule in algebra]:
$I = \frac{1}{2} (e - 1)$
Visual Representation of the Area under the Curve
Final Solution: \frac{e-1}{2}
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. $\int_{0}^{1} \frac{2x+3}{5x^2+1} dx$
- 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. $\int_{0}^{2} \frac{6x+3}{x^2+4} dx$
- Q20: Evaluate the definite integrals in Exercises 1 to 20. 20. $\int_{0}^{1} (x e^x + \sin \frac{\pi x}{4}) dx$
- 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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