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Q18:
Integrate the functions in Exercises 1 to 22.
18. $e^x (\frac{1+\sin x}{1+\cos x})$
Solution :
Step 1: Analyzing the Integrand
The given integral is: $I = \int e^x \left( \frac{1 + \sin x}{1 + \cos x} \right) \, dx$ [Per the general form $\int e^x [f(x) + f'(x)] \, dx = e^x f(x) + C$]
Step 2: Trigonometric Simplification
We simplify the expression $\frac{1 + \sin x}{1 + \cos x}$ using half-angle identities:
- $1 + \cos x = 2 \cos^2 \left( \frac{x}{2} \right)$ [Using $\cos 2\theta = 2\cos^2 \theta - 1$]
- $\sin x = 2 \sin \left( \frac{x}{2} \right) \cos \left( \frac{x}{2} \right)$ [Using $\sin 2\theta = 2 \sin \theta \cos \theta$]
Substituting these into the expression:
$f(x) = \frac{1 + 2 \sin(x/2) \cos(x/2)}{2 \cos^2(x/2)}$Step 3: Algebraic Decomposition
Divide the terms in the numerator by the denominator:
$f(x) = \frac{1}{2 \cos^2(x/2)} + \frac{2 \sin(x/2) \cos(x/2)}{2 \cos^2(x/2)}$ $f(x) = \frac{1}{2} \sec^2\left( \frac{x}{2} \right) + \tan\left( \frac{x}{2} \right)$Step 4: Identifying the Function and its Derivative
Let $f(x) = \tan\left( \frac{x}{2} \right)$.
Then, the derivative $f'(x)$ is:
$f'(x) = \frac{d}{dx} \left[ \tan\left( \frac{x}{2} \right) \right] = \sec^2\left( \frac{x}{2} \right) \cdot \frac{1}{2}$ $f'(x) = \frac{1}{2} \sec^2\left( \frac{x}{2} \right)$This matches our decomposed term, confirming the structure $e^x [f(x) + f'(x)]$.
Step 5: Applying the Integral Formula
Using the standard integration result for exponential-trigonometric products:
$\int e^x [f(x) + f'(x)] \, dx = e^x f(x) + C$Substituting $f(x) = \tan\left( \frac{x}{2} \right)$:
$I = e^x \tan\left( \frac{x}{2} \right) + C$Final Solution: $e^x \tan\left( \frac{x}{2} \right) + C$
More Questions from Class 12 Mathematics Integrals EXERCISE 7.6
- Q1: Integrate the functions in Exercises 1 to 22. 1. $x \sin x$
- Q2: Integrate the functions in Exercises 1 to 22. 2. $x \sin 3x$
- Q3: Integrate the functions in Exercises 1 to 22. 3. $x^2 e^x$
- Q4: Integrate the functions in Exercises 1 to 22. 4. $x \log x$
- Q5: Integrate the functions in Exercises 1 to 22. 5. $x \log 2x$
- Q6: Integrate the functions in Exercises 1 to 22. 6. $x^2 \log x$
- Q7: Integrate the functions in Exercises 1 to 22. 7. $x \sin^{-1} x$
- Q8: Integrate the functions in Exercises 1 to 22. 8. $x \tan^{-1} x$
- Q9: Integrate the functions in Exercises 1 to 22. 9. $x \cos^{-1} x$
- Q10: Integrate the functions in Exercises 1 to 22. 10. $(\sin^{-1} x)^2$
- Q11: Integrate the functions in Exercises 1 to 22. 11. $\frac{x \cos^{-1} x}{\sqrt{1-x^2}}$
- Q12: Integrate the functions in Exercises 1 to 22. 12. $x \sec^2 x$
- Q13: Integrate the functions in Exercises 1 to 22. 13. $\tan^{-1} x$
- Q14: Integrate the functions in Exercises 1 to 22. 14. $x (\log x)^2$
- Q15: Integrate the functions in Exercises 1 to 22. 15. $(x^2+1) \log x$
- Q16: Integrate the functions in Exercises 1 to 22. 16. $e^x(\sin x + \cos x)$
- Q17: Integrate the functions in Exercises 1 to 22. 17. $\frac{x e^x}{(1+x)^2}$
- Q19: Integrate the functions in Exercises 1 to 22. 19. $e^x (\frac{1}{x} - \frac{1}{x^2})$
- Q20: Integrate the functions in Exercises 1 to 22. 20. $\frac{(x-3)e^x}{(x-1)^3}$
- Q21: Integrate the functions in Exercises 1 to 22. 21. $e^{2x} \sin x$
- Q22: Integrate the functions in Exercises 1 to 22. 22. $\sin^{-1} (\frac{2x}{1+x^2})$
- Q23: Choose the correct answer in Exercises 23 and 24. 23. $\int x^2 e^{x^3} dx$ equals
- Q24: Choose the correct answer in Exercises 23 and 24. 24. $\int e^x \sec x (1 + \tan x) dx$ 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 7.7
- Integrals Class 12 EXERCISE 7.8
- Integrals Class 12 EXERCISE 7.9
- Integrals Class 12 Miscellaneous Exercise on Chapter 7
Chapters in CBSE - Class 12 Mathematics
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