Question
Download Solution PDFWhat is \(\displaystyle\int_0^1 \rm xe^x\ dx\) equal to?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
Integration by parts: Integration by parts is a method to find integrals of products
The formula for integrating by parts is given by;
⇒ ∫ uv dx = u(x) ∫ v(x) dx - ∫ [u'(x) ∫ v(x) dx] dx
Where u is the function u(x) and v is the function v(x), and is chosen by following ILATE rule.
ILATE Rule: Usually, the preference order of this rule is based on some functions such as Inverse, Logarithm, Algebraic, Trigonometric and Exponent.
Calculation:
Given:
Let I = \(\displaystyle∫_0^1 \rm xe^x\ dx\)
Applying by parts rule, we get
I \(\rm = x∫ _0^1 e^{x}\;dx - ∫ _0^1(\frac{dx}{dx}∫ e^{x}\;dx)\;dx \)
\(I\rm = [xe^{x}]_{0}^{1} - ∫ _0^11\times e^{x}\;dx\)
\(I\rm = [xe^{x}]_{0}^{1} - [e^{x}]_{0}^{1} \)
I = (e - 0) - (e - 1)
I = 1
Hence the required value of integration is 1.
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