Correct Answer is : e2x4(2x4−4x3+6x2−6x+3)+C{e²x}}{4}(2x^4 - 4x³ + 6x² - 6x + 3) + C4e2x(2x4−4x3+6x2−6x+3)+C

{e²x}}{4}(2x^4 - 4x³ + 6x² - 6x + 3) + C

{e²x}}{2}(2x^4 - 4x³ + 6x² - 6x + 3) + C

{e²x}}{8}(2x^4 + 4x³ + 6x² - 6x + 3) + C

-{e²x}}{4}(2x^4 + 4x³ + 6x² + 6x + 3) + C

Correct Answer is : e2x4(2x4−4x3+6x2−6x+3)+C{e²x}}{4}(2x^4 - 4x³ + 6x² - 6x + 3) + C4e2x(2x4−4x3+6x2−6x+3)+C

Test Index

TS EAMCET 2017 Code A Solved Paper

Section: Mathematics

Question : 20 of 160

Marks: +1, -0

\int x^4 e^{2x} dx =

Solution:

Solve the integral as follows.

I = \int x^{4} e^{2x} d x = x^{4} \int e^{2x} d x - \int \; \; \frac{d x^{4}}{d x} \int e^{2x} d x \cdot d x + C

= \; \; \frac{x^{4} e^{2x}}{2} - \; \; \frac{1}{2} \int 4 x^{3} e^{2x} d x + C

= \; \; \frac{x^{4} e^{2x}}{2} - 2\left[ \; \; \frac{x^{3} e^{2x}}{2} - \int \; \; \frac{3 x^{2} e^{2x}}{2} d x\right] + C

= \; \; \frac{x^{4} e^{2x}}{2} - x^{3} e^{2x} + 3\left[ \; \; \frac{x^{2} e^{2x}}{2} - \; \; \frac{1}{2} \int 2 x e^{2x} d x\right] + C

= \; \; \frac{x^{4} e^{2x}}{2} - x^{3} e^{2x} + \; \; \frac{3}{2} x^{2} e^{2x} - 3\left[ \; \; \frac{x e^{2x}}{2} - \int \; \; \frac{e^{2x}}{2} d x\right] + C

= \; \; \frac{x^{4} e^{2x}}{2} - x^{3} e^{2x} + \; \; \frac{3}{2} x^{2} e^{2x} - \; \; \frac{3}{2} x e^{2x} + \; \; \frac{3 e^{2x}}{4} + C

= \; \; \frac{e^{2x}}{4}\left[2 x^{4} - 4 x^{3} + 6 x^{2} - 6 x + 3\right] + c

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