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Definite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDefinite Integrals1 Mark
Question
The value of $\int\limits^\frac{\pi}{4}_0\cos\text{x}\text{ e}^{\sin\text{x}}\text{ dx}$ is:
  1. 1
  2. e - 1
  3. 0
  4. 1
✓

Answer

  1. $\text{e}-1$

Solution:

Let, $\text{I}=\int\limits^\frac{\pi}{4}_0\cos\text{x}\text{ e}^{\sin\text{x}}\text{dx}$

Let $\sin\text{x}=\text{t},$ then $\cos\text{x}\text{ dx}=\text{dt}$

when $\text{x}=0,\text{t}=0$ and $\text{x}=\frac{\pi}{2},\text{t}=1$

Therefore the integrel becomes

$\text{I}=\int\limits^1_0\text{e}^\text{t}\text{dt}$

$=\big[\text{e}^\text{t}\big]^1_0$

$=\text{e}-1$

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