We could not solve this problem by using the method: Integrate by parts
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We can solve the integral $\int\frac{x^2}{\sqrt{x^2+6}}dx$ 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 $\sqrt{x^2+6}$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
$u=\sqrt{x^2+6}$
Intermediate steps
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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
Substituting $u$, $dx$ and $x$ in the integral and simplify
$\int\sqrt{u^{2}-6}du$
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We can solve the integral $\int\sqrt{u^{2}-6}du$ by applying integration method of trigonometric substitution using the substitution
$u=\sqrt{6}\sec\left(\theta \right)$
Intermediate steps
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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
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 $\frac{1}{2}$
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
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 $\frac{6}{\log^{3}\left(10\right)}\int\sec\left(\theta \right)^{3}d\theta$ results in: $\frac{1}{2}\sqrt{x^2+6}x+3\ln\left(\frac{89}{218}\sqrt{x^2+6}+\frac{89}{218}x\right)$
The integral $\frac{6}{\log^{3}\left(10\right)}\int-\sec\left(\theta \right)d\theta$ results in: $-\frac{6}{\log^{3}\left(10\right)}\ln\left(\frac{89}{218}u+\frac{\sqrt{u^{2}-6}}{\sqrt{6}}\right)$
Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more
Given a function f(x) and the interval [a,b], the definite integral is equal to the area that is bounded by the graph of f(x), the x-axis and the vertical lines x=a and x=b