AP EAMCET · Maths · Indefinite Integration
\(\int e^{2 x}\left[\cos (3 x+4)+5 x^2\right] d x=\)
- A \(e^{2 x}\left[\frac{2}{13} \cos (3 x+4)+\frac{3}{13} \sin (3 x+4)+\frac{5 x^2}{2}-\frac{5 x}{2}+\frac{5}{4}\right]\)
- B \(e^{2 x}\left[\frac{2}{13} \cos (3 x+4)-\frac{3}{13} \sin (3 x+4)+\frac{5 x^2}{2}+\frac{5 x}{2}+\frac{5}{4}\right]^{+}\)
- C \(e^{2 x}\left[\frac{2}{13} \cos (3 x+4)-\frac{3}{13} \sin (3 x+4)-\frac{5 x^2}{2}-\frac{5 x}{2}-\frac{5}{4}\right]+c\)
- D \(e^{2 x}\left[\frac{2}{13} \cos (3 x+4)-\frac{3}{13} \sin (3 x+4)+\frac{5 x^2}{2}-\frac{5 x}{2}+\frac{5}{4}\right]\)
Answer & Solution
Correct Answer
(A) \(e^{2 x}\left[\frac{2}{13} \cos (3 x+4)+\frac{3}{13} \sin (3 x+4)+\frac{5 x^2}{2}-\frac{5 x}{2}+\frac{5}{4}\right]\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & \int e^{2 x}\left[\cos (3 x+4)+5 x^2\right] d x \\ & =\int e^{2 x} \cos (3 x+4) d x+5 \int e^{2 x} x^2 d x \\ & \left[\because \int e^{a x} \cos (b x+c) d x=\frac{e^{a x}}{a^2+b^2}(a \cos (b x+c)+b \sin (b x+c))+c\right] \end{aligned}\)…
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