∫ dx/(1+sqrt(x)) શોધો.
Understand the Problem
આ પ્રશ્ન આપણને ∫ dx/(1+sqrt(x)) નું સંકલન શોધવાનું કહે છે. આનું સમાકલન શોધવા માટે, આપણે યોગ્ય પરીવર્તનનો ઉપયોગ કરવાની જરૂર છે.
Answer
$2\sqrt{x} - 2\ln|1+\sqrt{x}| + C$
Answer for screen readers
$ 2\sqrt{x} - 2\ln|1+\sqrt{x}| + C $
Steps to Solve
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Substitution Let's use the substitution $u = \sqrt{x}$. This implies $x = u^2$. Then, differentiating both sides with respect to $u$, we get $dx = 2u , du$.
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Rewrite the integral Substituting $u$ and $dx$ into the integral, we get: $$ \int \frac{dx}{1 + \sqrt{x}} = \int \frac{2u}{1+u} , du $$
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Simplify the integral Now, we can rewrite the numerator to simplify the integral: $$ \int \frac{2u}{1+u} , du = 2 \int \frac{u}{1+u} , du = 2 \int \frac{u+1-1}{1+u} , du = 2 \int \left( 1 - \frac{1}{1+u} \right) , du $$
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Split the integral Splitting the integral, we have $$ 2 \int \left( 1 - \frac{1}{1+u} \right) , du = 2 \left( \int 1 , du - \int \frac{1}{1+u} , du \right) $$
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Integrate Now, we integrate each part: $$ 2 \left( \int 1 , du - \int \frac{1}{1+u} , du \right) = 2 \left( u - \ln|1+u| \right) + C = 2u - 2\ln|1+u| + C $$
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Substitute back Finally, substitute $u = \sqrt{x}$ back into the equation: $$ 2\sqrt{x} - 2\ln|1+\sqrt{x}| + C $$
$ 2\sqrt{x} - 2\ln|1+\sqrt{x}| + C $
More Information
The indefinite integral of $\frac{1}{1+\sqrt{x}}$ is $2\sqrt{x}-2\ln(1+\sqrt{x})+C$. Because the domain of the original function is $x \ge 0$, we know that $1 + \sqrt{x}$ is always positive, so the absolute value can be dropped.
Tips
A common mistake is forgetting to substitute back to the original variable after integration. Another common mistake is incorrectly simplifying the integral after substitution. Forgetting the constant of integration is also a common mistake.
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