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Question Number 73668 by liki last updated on 14/Nov/19

$$Qn.\int \left(e^{2x^2+2x+6}\right)dx$$$$...Ineedhelpplz..$$$$$$$$$$

Answered by arkanmath7@gmail.com last updated on 14/Nov/19

$$=\int e^{2x^2+2x}.e^{6}dx$$$$=e^6\int e^{2x^2+2x}dx$$$$=e^6\int e^{2x^2+2x+\frac{1}{2}-\frac{1}{2}}dx$$$$=e^6\int e^{{(\sqrt{2}x+\frac{1}{\sqrt{2}})}^2-\frac{1}{2}}dx$$$$=e^{-\frac{1}{2}}{e}^6\int e^{{\left(\frac{2x+1}{\sqrt{2}}\right)}^2}dx$$$$$$$$nowletu=\frac{2x+1}{\sqrt{2}}\Rightarrow du=\sqrt{2}dx$$$$\Rightarrow du=\sqrt{2}dx\Rightarrow dx=\frac{1}{\sqrt{2}}du$$$$\int e^{2x^2+2x+6}dx=e^{\frac{11}{2}}.\frac{1}{\sqrt{2}}\int e^{u^2}du$$$$=\frac{\sqrt{\pi }{e}^{\frac{11}{2}}}{2\sqrt{2}}\int \frac{2e^{u^2}}{\sqrt{\pi }}duerrorfunction$$$$=\frac{\sqrt{\pi }{e}^{\frac{11}{2}}}{2\sqrt{2}}erf\left(u\right)$$$$=\frac{\sqrt{\pi }{e}^{\frac{11}{2}}}{2\sqrt{2}}erf\left(\frac{2x+1}{\sqrt{2}}\right)+c$$$$$$$$$$$$$$$$$$$$$$$$$$

Commented by liki last updated on 14/Nov/19

$$Thankssomuch..$$

Commented by liki last updated on 14/Nov/19

well soln

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