We can solve the integral $\int\frac{x^2}{\sqrt{x^2+6}}dx$ by applying integration method of trigonometric substitution using the substitution
$x=\sqrt{6}\tan\left(\theta \right)$
Intermediate steps
2
Now, in order to rewrite $d\theta$ in terms of $dx$, we need to find the derivative of $x$. We need to calculate $dx$, we can do that by deriving the equation above
Simplify $\sqrt{\sec\left(\theta \right)^2}$ 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 $\frac{1}{2}$
Expand the fraction $\frac{1-\cos\left(\theta \right)^2}{\cos\left(\theta \right)^{3}}$ into $2$ simpler fractions with common denominator $\cos\left(\theta \right)^{3}$
Expand the integral $\int\left(\frac{1}{\cos\left(\theta \right)^{3}}+\frac{-1}{\cos\left(\theta \right)}\right)d\theta$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately
The integral $6\int\frac{1}{\cos\left(\theta \right)^{3}}d\theta$ results in: $\frac{1}{2}\sqrt{x^2+6}x+3\ln\left(\frac{\sqrt{x^2+6}}{\sqrt{6}}+\frac{x}{\sqrt{6}}\right)$
Combining like terms $3\ln\left(\frac{\sqrt{x^2+6}}{\sqrt{6}}+\frac{x}{\sqrt{6}}\right)$ and $-6\ln\left(\frac{\sqrt{x^2+6}}{\sqrt{6}}+\frac{x}{\sqrt{6}}\right)$
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
$L.C.M.=\sqrt{6}$
23
Combine and simplify all terms in the same fraction with common denominator $\sqrt{6}$
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