Q70 / 180 · COMEDK 2026 (9 May Shift 1) · Mathematics
$\int e^{2x}\cos(5x + 3)dx =$
- A$\dfrac{e^{2x}}{29}[5\cos(5x+3) - 2\sin(5x+3)] + C$
- B$\dfrac{e^{2x}}{29}[2\cos(5x+3) - 5\sin(5x+3)] + C$
- Correct answerC$\dfrac{e^{2x}}{29}[2\cos(5x+3) + 5\sin(5x+3)] + C$
- D$\dfrac{e^{2x}}{29}[2\sin(5x+3) + 5\cos(5x+3)] + C$
Solution
Using the standard integral formula:
$\int e^{ax}\cos(bx + c)dx = \dfrac{e^{ax}}{a^2 + b^2}[a\cos(bx + c) + b\sin(bx + c)] + C$
Comparing the given integral $\int e^{2x}\cos(5x + 3)dx$ with the standard form, we get $a = 2$, $b = 5$, and $c = 3$.
Substituting these values into the formula:
$\int e^{2x}\cos(5x + 3)dx = \dfrac{e^{2x}}{2^2 + 5^2}[2\cos(5x + 3) + 5\sin(5x + 3)] + C$
$= \dfrac{e^{2x}}{29}[2\cos(5x + 3) + 5\sin(5x + 3)] + C$
Answer: $\dfrac{e^{2x}}{29}[2\cos(5x+3) + 5\sin(5x+3)] + C$
$\int e^{ax}\cos(bx + c)dx = \dfrac{e^{ax}}{a^2 + b^2}[a\cos(bx + c) + b\sin(bx + c)] + C$
Comparing the given integral $\int e^{2x}\cos(5x + 3)dx$ with the standard form, we get $a = 2$, $b = 5$, and $c = 3$.
Substituting these values into the formula:
$\int e^{2x}\cos(5x + 3)dx = \dfrac{e^{2x}}{2^2 + 5^2}[2\cos(5x + 3) + 5\sin(5x + 3)] + C$
$= \dfrac{e^{2x}}{29}[2\cos(5x + 3) + 5\sin(5x + 3)] + C$
Answer: $\dfrac{e^{2x}}{29}[2\cos(5x+3) + 5\sin(5x+3)] + C$
Hence, the correct answer is option: 'C'.