∫ x²≤ft[{2}≤ft(π/4+x)+e^x] dx =

Correct Answer is : (x2+2x−2)(x²+2x-2)(x2+2x−2) sin x + (−x2+2x+2)(-x²+2x+2)(−x2+2x+2) cos x + (x2−2x+2)ex(x²-2x+2)e^x(x2−2x+2)ex + c

(x²+2x-2) sin x + (-x²+2x+2) cos x + (x²-2x+2)e^x + c

(-x²+2x-2) sin x + (x²+2x+2) cos x + (x²-2x+2)e^x + c

(x²+2x+2) sin x + (-x²-2x-2) cos x + (x²-2x+2)e^x + c

(x²-2x-2) sin x + (-x²+2x-2) cos x + (x²-2x+2)e^x + c

Correct Answer is : (x2+2x−2)(x²+2x-2)(x2+2x−2) sin x + (−x2+2x+2)(-x²+2x+2)(−x2+2x+2) cos x + (x2−2x+2)ex(x²-2x+2)e^x(x2−2x+2)ex + c

Test Index

AP EAMCET Engineering 21 Apr 2019 Shift 2 Paper

Section: Mathematics

Question : 73 of 160

Marks: +1, -0

∫

x^2\left[\sqrt{2}\sin\left(\frac{\pi}{4}+x\right)+e^x\right]

dx =

Solution:

The integral expression is given as,

I=\int x^{2}\left[\sqrt{2}\left(\sin\frac{\pi}{4}+x\right)+e^{x}\right]\,dx

I=\int x^{2}\left[\sqrt{2}\left(\sin\frac{\pi}{4}\cos x+\sin x\cos\frac{\pi}{4}\right)+e^{x}\right]\,dx

I=\int x^{2}\left[\sqrt{2}\left(\frac{1}{\sqrt{2}}\cos x+\frac{1}{\sqrt{2}}\sin x\right)+e^{x}\right]\,dx

Further simplify,

I=\int x^{2}[\cos x+\sin x+e^{x}]\,dx

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

I=\left\{\begin{array}{c}(x^{2}+2x-2)\sin x-\\(x^{2}-2x-2)\cos x\end{array}\right\}+\{x^{2}e^{x}-2xe^{x}+2e^{x}\}+c

I=(x^{2}+2x-2)\sin x+(-x^{2}+2x+2)\cos x+(x^{2}-2x+2)e^{x}+c

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