Final Answer
Step-by-step Solution
Specify the solving method
We can solve the integral $\int\ln\left(\sqrt{x}+\sqrt{1+x}\right)dx$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula
The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$
The derivative of a sum of two or more functions is the sum of the derivatives of each function
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
The derivative of a sum of two or more functions is the sum of the derivatives of each function
The derivative of the constant function ($1$) is equal to zero
The derivative of the linear function is equal to $1$
Multiplying the fraction by $\frac{1}{2}x^{-\frac{1}{2}}+\frac{1}{2}\left(1+x\right)^{-\frac{1}{2}}$
Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number
The least common multiple (LCM) of a sum of algebraic fractions consists of the product of the common factors with the greatest exponent, and the uncommon factors
Obtained the least common multiple (LCM), we place it as the denominator of each fraction, and in the numerator of each fraction we add the factors that we need to complete
Combine and simplify all terms in the same fraction with common denominator $2\sqrt{x}\sqrt{1+x}$
Divide fractions $\frac{\frac{\sqrt{1+x}+\sqrt{x}}{2\sqrt{x}\sqrt{1+x}}}{\sqrt{x}+\sqrt{1+x}}$ with Keep, Change, Flip: $\frac{a}{b}\div c=\frac{a}{b}\div\frac{c}{1}=\frac{a}{b}\times\frac{1}{c}=\frac{a}{b\cdot c}$
Simplify the fraction $\frac{\sqrt{1+x}+\sqrt{x}}{2\sqrt{x}\sqrt{1+x}\left(\sqrt{x}+\sqrt{1+x}\right)}$ by $\sqrt{1+x}+\sqrt{x}$
First, identify $u$ and calculate $du$
Now, identify $dv$ and calculate $v$
Solve the integral
The integral of a constant is equal to the constant times the integral's variable
Simplify the fraction by $x$
Now replace the values of $u$, $du$ and $v$ in the last formula
Take the constant $\frac{1}{2}$ out of the integral
Multiply $-1$ times $\frac{1}{2}$
We can solve the integral $\int\frac{\sqrt{x}}{\sqrt{1+x}}dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $\sqrt{1+x}$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
Now, in order to rewrite $dx$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above
Isolate $dx$ in the previous equation
Rewriting $x$ in terms of $u$
Substituting $u$, $dx$ and $x$ in the integral and simplify
The integral of a function times a constant ($2$) is equal to the constant times the integral of the function
We can solve the integral $-\int\sqrt{u^{2}-1}du$ by applying integration method of trigonometric substitution using the substitution
Now, in order to rewrite $d\theta$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above
Substituting in the original integral, we get
Apply the trigonometric identity: $\sec\left(\theta \right)^2-1$$=\tan\left(\theta \right)^2$, where $x=\theta $
Simplify $\sqrt{\tan\left(\theta \right)^2}$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $2$ and $n$ equals $\frac{1}{2}$
When multiplying two powers that have the same base ($\tan\left(\theta \right)$), you can add the exponents
We identify that the integral has the form $\int\tan^m(x)\sec^n(x)dx$. If $n$ is odd and $m$ is even, then we need to express everything in terms of secant, expand and integrate each function separately
Multiply the single term $\sec\left(\theta \right)$ by each term of the polynomial $\left(\sec\left(\theta \right)^2-1\right)$
Expand the integral $\int\left(\sec\left(\theta \right)^{3}-\sec\left(\theta \right)\right)d\theta$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately
The integral of a function times a constant ($-1$) is equal to the constant times the integral of the function
Any expression multiplied by $1$ is equal to itself
The integral of the secant function is given by the following formula, $\displaystyle\int\sec(x)dx=\ln\left|\sec(x)+\tan(x)\right|$
Express the variable $\theta$ in terms of the original variable $x$
Replace $u$ with the value that we assigned to it in the beginning: $\sqrt{1+x}$
Subtract the values $1$ and $-1$
Rewrite $\sec\left(\theta \right)^{3}$ as the product of two secants
We can solve the integral $\int\sec\left(\theta \right)^2\sec\left(\theta \right)d\theta$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula
First, identify $u$ and calculate $du$
Now, identify $dv$ and calculate $v$
Solve the integral
The integral of $\sec(x)^2$ is $\tan(x)$
Now replace the values of $u$, $du$ and $v$ in the last formula
Multiply the single term $-1$ by each term of the polynomial $\left(\tan\left(\theta \right)\sec\left(\theta \right)-\int\tan\left(\theta \right)^2\sec\left(\theta \right)d\theta\right)$
Express the variable $\theta$ in terms of the original variable $x$
Replace $u$ with the value that we assigned to it in the beginning: $\sqrt{1+x}$
Subtract the values $1$ and $-1$
Apply the formula: $\int\sec\left(\theta \right)\tan\left(\theta \right)^2dx$$=\int\sec\left(\theta \right)^3dx-\int\sec\left(\theta \right)dx$, where $x=\theta $
The integral of the secant function is given by the following formula, $\displaystyle\int\sec(x)dx=\ln\left|\sec(x)+\tan(x)\right|$
Express the variable $\theta$ in terms of the original variable $x$
Replace $u$ with the value that we assigned to it in the beginning: $\sqrt{1+x}$
Subtract the values $1$ and $-1$
Cancel like terms $-\ln\left(\sqrt{1+x}+\sqrt{x}\right)$ and $\ln\left(\sqrt{1+x}+\sqrt{x}\right)$
Simplify the integral $\int\sec\left(\theta \right)^3d\theta$ applying the reduction formula, $\displaystyle\int\sec(x)^{n}dx=\frac{\sin(x)\sec(x)^{n-1}}{n-1}+\frac{n-2}{n-1}\int\sec(x)^{n-2}dx$
Simplify the expression inside the integral
Express the variable $\theta$ in terms of the original variable $x$
Replace $u$ with the value that we assigned to it in the beginning: $\sqrt{1+x}$
Subtract the values $1$ and $-1$
Simplify the expression inside the integral
The integral of the secant function is given by the following formula, $\displaystyle\int\sec(x)dx=\ln\left|\sec(x)+\tan(x)\right|$
Express the variable $\theta$ in terms of the original variable $x$
Replace $u$ with the value that we assigned to it in the beginning: $\sqrt{1+x}$
Subtract the values $1$ and $-1$
The integral $-\int\frac{\sqrt{x}}{2\sqrt{1+x}}dx$ results in: $-\frac{1}{2}\sqrt{x}\sqrt{1+x}+\frac{1}{2}\ln\left(\sqrt{1+x}+\sqrt{x}\right)$
Gather the results of all integrals
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$