"Evaluate the integrals. ∫01x⋅2x2+1dx" was answered by our community

sodni3

Answered question

2021-10-25

Evaluate the integrals.

\int_{0}^{1} x \cdot 2^{x^{2} + 1} d x

gotovub

Skilled2021-10-26Added 98 answers

Given:

\int_{0}^{1} x \cdot 2^{x^{2} + 1} d x

Applying substitution method,
let

u = x^{2}

\Rightarrow d u = 2 x d x

\Rightarrow \frac{d y}{2} = x d x

The integral becomes

= \int_{0}^{1} \frac{1}{2} \cdot 2^{4 + 1} d u

= \int_{0}^{1} \frac{2}{2} \cdot 2^{4} d u

= \int_{0}^{1} 2^{4} d u

= {[\frac{2^{4}}{\ln (2)}]}_{0}^{1}

= \frac{2}{\ln (2)} - \frac{1}{n (2)}

= \frac{1}{\ln (2)}

∴ \int_{0}^{1} x \cdot 2^{x^{2} + 1} d x = \frac{1}{\ln (2)}

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