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Exercise

$\int\left(\frac{\sin^{-1}\left(x\right)}{x^3}\right)dx$

Step-by-step Solution

1

Rewrite the exponent using the power rule $\frac{a^m}{a^n}=a^{m-n}$, where in this case $m=0$

$\int x^{-3}\arcsin\left(x\right)dx$
2

We can solve the integral $\int x^{-3}\arcsin\left(x\right)dx$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula

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$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
3

First, identify or choose $u$ and calculate it's derivative, $du$

$\begin{matrix}\displaystyle{u=\arcsin\left(x\right)}\\ \displaystyle{du=\frac{1}{\sqrt{1-x^2}}dx}\end{matrix}$
4

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=x^{-3}dx}\\ \displaystyle{\int dv=\int x^{-3}dx}\end{matrix}$
5

Solve the integral to find $v$

$v=\int x^{-3}dx$
6

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, such as $-3$

$\frac{x^{-2}}{-2}$
7

Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number

$\frac{\frac{1}{x^{2}}}{-2}$
8

Now replace the values of $u$, $du$ and $v$ in the last formula

$\frac{\arcsin\left(x\right)}{-2x^{2}}-\int\frac{1}{-2\sqrt{1-x^2}x^{2}}dx$
9

The integral $-\int\frac{1}{-2\sqrt{1-x^2}x^{2}}dx$ results in: $\frac{-\sqrt{1-x^2}}{2x}$

$\frac{-\sqrt{1-x^2}}{2x}$
10

Gather the results of all integrals

$\frac{\arcsin\left(x\right)}{-2x^{2}}+\frac{-\sqrt{1-x^2}}{2x}$
11

Simplify the expression

$\frac{-\arcsin\left(x\right)-\sqrt{1-x^2}x}{2x^{2}}$
12

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

$\frac{-\arcsin\left(x\right)-\sqrt{1-x^2}x}{2x^{2}}+C_0$

Final answer to the exercise

$\frac{-\arcsin\left(x\right)-\sqrt{1-x^2}x}{2x^{2}}+C_0$

Try other ways to solve this exercise

  • Choose an option
  • Integrate by partial fractions
  • Integrate by substitution
  • Integrate by parts
  • Integrate using tabular integration
  • Integrate by trigonometric substitution
  • Weierstrass Substitution
  • Integrate using trigonometric identities
  • Integrate using basic integrals
  • Product of Binomials with Common Term
  • Load more...
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