If I 1 = ∫ 0 1 1 + x e x d x and I 2 = ∫ 0 1 e x 3 ( 2 − x 3 ) x 2 d x , find I 1 / I 2 .

Put z = 1 − x 3 in I 2 . Then I 2 = 3 1 ∫ 0 1 1 + z e z − 1 d z = I 1 / ( 3 e ) . Hence I 1 / I 2 = 3 e .

Mathematics - Definite Integration Question with Solution | TestHub

MathematicsDefinite IntegrationDefinite Integration by SubstitutionHard2 minQB

MathematicsHardsingle choice

If $I_{1} = \int_{0}^{1} \frac{e ^{x}}{1 + x} d x$ and $I_{2} = \int_{0}^{1} \frac{x ^{2}}{e ^{x^{3}} ( 2 - x ^{3} )} d x$ , find $I_{1} / I_{2}$ .

Answer:

Solution:

Put $z = 1 - x^{3}$ in $I_{2}$ . Then $I_{2} = \frac{1}{3} \int_{0}^{1} \frac{e ^{z - 1}}{1 + z} d z = I_{1} / (3 e)$ . Hence $I_{1} / I_{2} = 3 e$ .

Stream:JEESubject:MathematicsTopic:Definite IntegrationSubtopic:Definite Integration by Substitution

⏱ 2mℹ️ Source: QB

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