NCERT Class 12 · Ch 7 Exercise 7.1 · Q5
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.1 Question 5
Evaluate \(\int \left(- 4 e^{3 x} + \sin{\left(2 x \right)}\right)\, dx\).
💡 Concept Hints
- Identify the domain and the inner and outer functions, or the relevant one-sided limits.
- Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.
✅ Step-by-Step Solution
Final Answer: \(- \frac{4 e^{3 x}}{3} - \frac{\cos{\left(2 x \right)}}{2}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Expand and use linearity, then integrate each power or elementary term.
- The integrand rewrites as \(- 4 e^{3 x} + \sin{\left(2 x \right)}\). Apply ∫xᵖdx=x^(p+1)/(p+1) for p≠−1, ∫dx/x=log|x|, and the elementary exponential/trigonometric rules.
- An antiderivative is F(x)=\(- \frac{4 e^{3 x}}{3} - \frac{\cos{\left(2 x \right)}}{2}\).
- Differentiate this expression to check that F′(x)=\(- 4 e^{3 x} + \sin{\left(2 x \right)}\). Add an arbitrary constant C on each domain component.
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