NCERT Class 12 · Ch 7 Exercise 7.6 · Q21
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.6 Question 21
Evaluate \(\int e^{2 x} \sin{\left(x \right)}\, dx\).
💡 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: \(\frac{\left(2 \sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{2 x}}{5}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Apply integration by parts twice, choosing the trigonometric factor as u.
- Writing I=∫e^(2x)sin x dx yields I=e^(2x)sin x/2−e^(2x)cos x/4−I/4. Collect (5/4)I and solve.
- An antiderivative is F(x)=\(\frac{\left(2 \sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{2 x}}{5}\).
- Differentiate this expression to check that F′(x)=\(e^{2 x} \sin{\left(x \right)}\). Add an arbitrary constant C on each domain component.
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