∫ e²x} [cos (3x + 4) + 5x²] dx =

Correct Answer is : e2xe²x}e2x [213cos⁡(3x+4)+313sin⁡(3x+4)+5x22−5x2+54]≤ft[ 2/13 (3x+4) + 3/13 (3x+4) + 5x²/2 - 5x/2 + 5/4 ][132cos(3x+4)+133sin(3x+4)+25x2−25x+45] + c

e²x} ≤ft[ 2/13 (3x+4) + 3/13 (3x+4) + 5x²/2 - 5x/2 + 5/4 ] + c

e²x} ≤ft[ 2/13 (3x+4) - 3/13 (3x+4) + 5x²/2 + 5x/2 + 5/4 ] + c

e²x} ≤ft[ 2/13 (3x+4) + 3/13 (3x+4) + 5x²/2 - 5x/2 - 5/4 ] + c

e²x} ≤ft[ 2/13 (3x+4) - 3/13 (3x+4) + 5x²/2 - 5x/2 + 5/4 ] + c

∫ e²x} [cos (3x + 4) + 5x²] dx =

Correct Answer is : e2xe²x}e2x [213cos⁡(3x+4)+313sin⁡(3x+4)+5x22−5x2+54]≤ft[ 2/13 (3x+4) + 3/13 (3x+4) + 5x²/2 - 5x/2 + 5/4 ][132cos(3x+4)+133sin(3x+4)+25x2−25x+45] + c

Test Index

AP EAMCET Engineering 23 Apr 2019 Shift 1 Paper

Section: Mathematics

Question : 73 of 160

Marks: +1, -0

∫

e^{2x}

[cos (3x + 4) +

5x^2

] dx =

$e^{2x}$
$\left[ \frac{2}{13} \cos(3x+4) + \frac{3}{13} \sin(3x+4) + \frac{5x^2}{2} - \frac{5x}{2} + \frac{5}{4} \right]$
+ c
$e^{2x}$
$\left[ \frac{2}{13} \cos(3x+4) - \frac{3}{13} \sin(3x+4) + \frac{5x^2}{2} + \frac{5x}{2} + \frac{5}{4} \right]$
+ c
$e^{2x}$
$\left[ \frac{2}{13} \cos(3x+4) + \frac{3}{13} \sin(3x+4) + \frac{5x^2}{2} - \frac{5x}{2} - \frac{5}{4} \right]$
+ c
$e^{2x}$
$\left[ \frac{2}{13} \cos(3x+4) - \frac{3}{13} \sin(3x+4) + \frac{5x^2}{2} - \frac{5x}{2} + \frac{5}{4} \right]$
+ c

Validate

Solution:

Consider the integral,

I(x)=\int e^{2x}(\cos(3x+4)+5x^2)\,dx

=\int e^{2x}\cos(3x+4)\,dx + 5\int e^{2x} x^2\,dx

= e^{2x} \left[ \frac{2}{13} \cos(3x+4) + \frac{3}{13} \sin(3x+4) \right]

+ \frac{5x^2}{2} e^{2x} - 5\int x e^{2x}\,dx

= e^{2x} \left( \frac{2}{13} \cos(3x+4) + \frac{3}{13} \sin(3x+4) \right)

+ \frac{5x^2}{2} e^{2x} - \frac{5}{2} x e^{2x} + \frac{5}{2} \int e^{2x}\,dx

Further simplify the above,

= e^{2x} \left( \frac{2}{13} \cos(3x+4) + \frac{3}{13} \sin(3x+4) \right)

+ \frac{5x^2}{2} e^{2x} - \frac{5}{2} x e^{2x} + \frac{5}{4} e^{2x}

e^{2x} \left[ \frac{2}{13} \cos(3x+4) + \frac{3}{13} \sin(3x+4) + \frac{5x^2}{2} - \frac{5x}{2} + \frac{5}{4} \right]

+ c

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