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Q11:
Find the following integrals 11. $\int \frac{x^3 + 5x^2 - 4}{x^2} dx$
Find the following integrals 11. $\int \frac{x^3 + 5x^2 - 4}{x^2} dx$
Solution :
Step 1: Decomposition of the Rational Function
Given the integral: $I = \int \frac{x^3 + 5x^2 - 4}{x^2} dx$
To evaluate this, we distribute the denominator $x^2$ to each term in the numerator. This is permissible as $x \neq 0$ [By the property of algebraic division of polynomials]:
$I = \int \left( \frac{x^3}{x^2} + \frac{5x^2}{x^2} - \frac{4}{x^2} \right) dx$
Step 2: Simplification of Terms
Apply the laws of exponents $\frac{x^a}{x^b} = x^{a-b}$ and $\frac{1}{x^n} = x^{-n}$ to simplify the integrand:
$I = \int \left( x + 5 - 4x^{-2} \right) dx$
Step 3: Applying Linearity of Integration
The integral of a sum/difference is the sum/difference of the integrals [Per the Sum/Difference Rule of Integration]. We split the integral into individual components:
$I = \int x \, dx + \int 5 \, dx - \int 4x^{-2} \, dx$
Step 4: Integration Using Power Rule
Apply the Power Rule $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$:
- $\int x^1 dx = \frac{x^{1+1}}{1+1} = \frac{x^2}{2}$
- $\int 5 dx = 5x$
- $\int 4x^{-2} dx = 4 \left( \frac{x^{-2+1}}{-2+1} \right) = 4 \left( \frac{x^{-1}}{-1} \right) = -4x^{-1}$
Step 5: Final Assembly and Integration Constant
Combine the results and include the constant of integration $C$:
$I = \frac{x^2}{2} + 5x - (-4x^{-1}) + C$
$I = \frac{x^2}{2} + 5x + \frac{4}{x} + C$
Final Solution: $\frac{x^2}{2} + 5x + \frac{4}{x} + C$
More Questions from Class 12 Mathematics Integrals EXERCISE 7.1
- Q1: Find an anti derivative (or integral) of the following functions by the method of inspection. 1. $\sin 2x$
- Q2: Find an anti derivative (or integral) of the following functions by the method of inspection. 2. $\cos 3x$
- Q3: Find an anti derivative (or integral) of the following functions by the method of inspection. 3. $e^{2x}$
- Q4: Find an anti derivative (or integral) of the following functions by the method of inspection. 4. $(ax + b)^2$
- Q5: Find an anti derivative (or integral) of the following functions by the method of inspection. 5. $\sin 2x - 4e^{3x}$
- Q6: Find the following integrals in Exercises 6 to 20: 6. $\int (4e^{3x} + 1) dx$
- Q7: Find the following integrals in Exercises 6 to 20: 7. $\int x^2(1 - \frac{1}{x^2}) dx$
- Q8: Find the following integrals 8. $\int (ax^2 + bx + c) dx$
- Q9: Find the following integrals 9. $\int (2x^2 + e^x) dx$
- Q10: Find the following integrals 10. $\int (\sqrt{x} - \frac{1}{\sqrt{x}})^2 dx$
- Q12: Find the following integrals 12. $\int \frac{x^3 + 3x + 4}{\sqrt{x}} dx$
- Q13: Find the following integrals 13. $\int \frac{x^3 - x^2 + x - 1}{x - 1} dx$
- Q14: Find the following integrals 14. $\int (1 - x)\sqrt{x} dx$
- Q15: Find the following integrals 15. $\int \sqrt{x} (3x^2 + 2x + 3) dx$
- Q16: Find the following integrals 16. $\int (2x - 3\cos x + e^x) dx$
- Q17: Find the following integrals 17. $\int (2x^2 - 3\sin x + 5\sqrt{x}) dx$
- Q18: Find the following integrals 18. $\int \sec x (\sec x + \tan x) dx$
- Q19: Find the following integrals 19. $\int \frac{\sec^2 x}{\csc^2 x} dx$
- Q20: Find the following integrals 20. $\int \frac{2 - 3\sin x}{\cos^2 x} dx$
- Q21: Choose the correct answer 21. The anti derivative of $(\sqrt{x} + \frac{1}{\sqrt{x}})$ equals
- Q22: Choose the correct answer 22. If $\frac{d}{dx} f(x) = 4x^3 - \frac{3}{x^4}$ such that $f(2) = 0$. Then $f(x)$ is
CBSE Solutions for Class 12 Mathematics Integrals
- 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 7.8
- Integrals Class 12 EXERCISE 7.9
- Integrals Class 12 Miscellaneous Exercise on Chapter 7
Chapters in CBSE - Class 12 Mathematics
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