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Question Number 88859 by 242242864 last updated on 13/Apr/20

$$\int {\mathit{e}}^{\mathit{a}\mathit{x}}\cos \mathit{b}\mathit{x}\mathit{d}\mathit{x}$$$$\int {\mathit{x}}^2{\mathit{e}}^{2\mathit{x}}\mathit{l}\mathit{n}3{\mathit{x}}^2\mathit{d}\mathit{x}$$

Commented by john santu last updated on 13/Apr/20

$$1)I=\int e^{ax}\cos bxdx$$$$I=\frac{1}{b}{e}^{ax}\sin bx-\frac{1}{ab}\int e^{ax}\sin bxdx$$$$I=\frac{e^{ax}\sin bx}{b}-\frac{1}{ab}[-\frac{e^{ax}\cos bx}{b}+\frac{1}{ab}\int e^{ax}\cos bxdx]$$$$I=\frac{e^{ax}\sin bx}{b}+\frac{e^{ax}\cos bx}{{ab}^2}-\frac{1}{{\left(ab\right)}^2}\int e^{ax}\cos bxdx$$$$(1+\frac{1}{{\left(ab\right)}^2})I=\frac{e^{ax}(ab\sin bx+\cos bx)}{{ab}^2}$$$$I=\frac{a^2{b}^2{e}^{ax}(ab\sin bx+\cos bx)}{(1+{\left(ab\right)}^2)\left({ab}^2\right)}$$$$I=\frac{{ae}^{ax}(ab\sin bx+\cos bx)}{(1+{\left(ab\right)}^2)}$$

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