₀¹ ≤ft( x e^x + (π x)/4 ) dx - NCERT Class XII Mathematics ...

NCERT Class XII Mathematics Chapter - - Solutions

Exercise - 7.9
Evaluate the definite integrals in Exercises 176 to 195.

Question : 195 of 271

Marks: +1, -0

\int\limits_{0}^{1} \left( x e^x + \frac{\sin(\pi x)}{4} \right) dx

Solution:

Let

I = \int\limits_{0}^{1} \left( x e^x + \frac{\sin(\pi x)}{4} \right) dx

= \int\limits_{0}^{1} x e^x dx + \int\limits_{0}^{1} \frac{\sin(\pi x)}{4} dx

= \left| x e^x \right|_{0}^{1} - \int\limits_{0}^{1} \left( \frac{d}{dx} x \cdot \int e^x dx \right) dx - \left[ \frac{ \frac{\cos(\pi x)}{4} }{ \frac{\pi}{4} } \right]_{0}^{1}

= \left| x e^x \right|_{0}^{1} - \int\limits_{0}^{1} e^x dx - \frac{4}{\pi} \left[ \frac{\cos(\pi x)}{4} \right]_{0}^{1}

= \left[ x e^x - e^x \right]_{0}^{1} - \frac{4}{\pi} \left( \cos \frac{\pi}{4} - \cos 0 \right)

= (e^1 - 0) - (e - e^0) - \frac{4}{\pi} \left( \frac{1}{\sqrt{2}} - 1 \right)

= e - e + 1 - \frac{4}{\pi \sqrt{2}} + \frac{4}{\pi}

= 1 - \frac{4}{\pi \sqrt{2}} + \frac{4}{\pi}

Go to Question: