Integral Of X Sqrt 1 X 2: A Cleaner Substitution Path
Integral of $$x\sqrt{1-x^2}$$: a cleaner substitution path
The integral is $$\int x\sqrt{1-x^2}\,dx = -\frac{1}{3}(1-x^2)^{3/2}+C$$. The cleanest method is the u-substitution $$u=1-x^2$$, which turns the problem into a one-step power rule integral.
Why this substitution works
This integral is designed for a direct substitution because the inner expression $$1-x^2$$ appears inside the square root, and its derivative is proportional to the outside factor $$x$$. That pairing is the key pattern to look for in calculus problems of this type.
- Let $$u=1-x^2$$.
- Then $$du=-2x\,dx$$, so $$x\,dx=-\frac12 du$$.
- Rewrite the integral as $$-\frac12\int u^{1/2}\,du$$.
- Apply the power rule: $$-\frac12\cdot \frac{u^{3/2}}{3/2}+C$$.
- Simplify to $$-\frac13u^{3/2}+C$$.
- Substitute back $$u=1-x^2$$.
Step-by-step solution
- Start with $$\int x\sqrt{1-x^2}\,dx$$.
- Set $$u=1-x^2$$.
- Differentiate to get $$du=-2x\,dx$$.
- Replace $$x\,dx$$ with $$-\frac12du$$.
- Convert the square root to a power: $$\sqrt{u}=u^{1/2}$$.
- Integrate: $$-\frac12\int u^{1/2}du=-\frac12\cdot\frac{2}{3}u^{3/2}+C$$.
- Return to $$x$$: $$-\frac13(1-x^2)^{3/2}+C$$.
Worked example
If you differentiate the result, you recover the original integrand: $$\frac{d}{dx}\left[-\frac13(1-x^2)^{3/2}\right]=x\sqrt{1-x^2}$$. This verification step is useful because it confirms both the algebra and the substitution choice.
| Step | Expression | Purpose |
|---|---|---|
| Original integral | $$\int x\sqrt{1-x^2}\,dx$$ | Recognize the composite function. |
| Substitution | $$u=1-x^2$$ | Match the inner function. |
| Differential | $$du=-2x\,dx$$ | Replace the outside factor. |
| Integrated form | $$-\frac12\int u^{1/2}du$$ | Simplify to a power rule. |
| Final answer | $$-\frac13(1-x^2)^{3/2}+C$$ | Substitute back. |
Common mistakes
One frequent error is treating $$\sqrt{1-x^2}$$ as though it were $$\sqrt{1}-\sqrt{x^2}$$, which is not valid. Another common mistake is forgetting the negative sign from $$du=-2x\,dx$$, which changes the final answer.
"Look for an inner function and its derivative outside." That simple habit solves a large share of substitution problems in calculus.
FAQ
Everything you need to know about Integral Of X Sqrt 1 X 2 A Cleaner Substitution Path
What is the integral of $$x\sqrt{1-x^2}$$?
The antiderivative is $$-\frac13(1-x^2)^{3/2}+C$$.
Why choose $$u=1-x^2$$?
Because the derivative of $$1-x^2$$ is $$-2x$$, and the integral already contains an $$x\,dx$$ factor that can be absorbed into $$du$$.
Can this be solved with trigonometric substitution?
Yes, but it is unnecessary here. A direct $$u$$-substitution is shorter, cleaner, and less error-prone.
How do I check the result?
Differentiate $$-\frac13(1-x^2)^{3/2}$$. The derivative simplifies to $$x\sqrt{1-x^2}$$, confirming the answer.
