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Вычислить определённый интеграл

int_(0)^((3)/(4))(dx)/((x+1)sqrt(x^(2)+1))

$1$ . Вычислить определённый интеграл $\newline$ $\int_{0}^{\frac{3}{4}} \frac{d x}{(x+1) \sqrt{x^{2}+1}}$

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Evaluate definite integrals using the chain rule

Full solution

Q.

1

. Вычислить определённый интеграл

\newline

\int_{0}^{\frac{3}{4}} \frac{d x}{(x+1) \sqrt{x^{2}+1}}

Substitution: Let's do a substitution: let $u = x^2 + 1$ , then $du = 2x \, dx$ .
Rewrite in terms of $u$ : Rewrite the integral in terms of $u$ : $\int_{0}^{\frac{3}{4}}\frac{dx}{(x+1)\sqrt{x^{2}+1}}$ becomes $\frac{1}{2} \int_{1}^{\frac{13}{16}}\frac{du}{\sqrt{u}(u-1)}$ .
Split into two parts: Now, let's split the integral into two parts: (\(1/ $2$ ) \times \left(\int_{ $1$ }^{ $13$ / $16$ }\frac{du}{u^{ $3$ / $2$ }} - \int_{ $1$ }^{ $13$ / $16$ }\frac{du}{u^{ $1$ / $2$ }}\right)\.
Find antiderivatives: Find the antiderivatives: $(1/2) \cdot (-2/u^{1/2} - 2\sqrt{u})$ from $1$ to $13/16$ .

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