Integral Of 1 1 X 2 3 2 Solved With Clear Reasoning
Integral of 1 1 x 2 3 2 is most plausibly the typed form of $$\int \frac{1}{(1+x^2)^{3/2}}\,dx$$, whose antiderivative is $$\frac{x}{\sqrt{1+x^2}}+C$$.
What the expression means
The phrase typed notation matters here because the user input appears to be an OCR-style or spacing-stripped version of a standard calculus problem, not a literal arithmetic sentence. In clear mathematical form, the integrand is usually written with parentheses and an exponent: $$\frac{1}{(1+x^2)^{3/2}}$$.
This is a common indefinite integral from early calculus and trigonometric substitution practice, especially when the denominator has the form $$(1+x^2)^{3/2}$$.
How to solve it
- Recognize the standard form $$\int \frac{1}{(1+x^2)^{3/2}}\,dx$$.
- Use the substitution $$x=\tan\theta$$, so $$dx=\sec^2\theta\,d\theta$$.
- Rewrite $$1+x^2$$ as $$1+\tan^2\theta=\sec^2\theta$$, which simplifies the denominator.
- After simplification, the integral reduces to $$\int \cos\theta\,d\theta$$, which integrates to $$\sin\theta+C$$.
- Convert back to $$x$$ using a right triangle, giving $$\sin\theta=\frac{x}{\sqrt{1+x^2}}$$.
Result at a glance
| Expression | Antiderivative | Method |
|---|---|---|
| $$\int \frac{1}{(1+x^2)^{3/2}}\,dx$$ | $$\frac{x}{\sqrt{1+x^2}}+C$$ | Trig substitution |
Why this answer is correct
The derivative of $$\frac{x}{\sqrt{1+x^2}}$$ returns $$\frac{1}{(1+x^2)^{3/2}}$$, which confirms the antiderivative. That verification step is standard practice in calculus and is one of the fastest ways to check a proposed integral result.
Key rule: if the exponent pattern is $$3/2$$ on $$1+x^2$$, trig substitution is often the cleanest path rather than direct power rules.
Common mistakes
- Dropping the parentheses and misreading the integrand as separate terms.
- Forgetting the constant of integration $$C$$ in an indefinite integral.
- Trying to apply the basic power rule directly to $$(1+x^2)^{3/2}$$, which is not a simple $$x^n$$ term.
FAQ
"Integration is the opposite of differentiation basically," and in this case differentiating the proposed answer is the fastest way to confirm the result.
Practical takeaway
The most useful way to read integral notation problems is to restore the missing structure first: parentheses, exponents, and the $$dx$$ variable. Once the expression is reconstructed as $$\int \frac{1}{(1+x^2)^{3/2}}\,dx$$, the answer follows cleanly as $$\frac{x}{\sqrt{1+x^2}}+C$$.
Everything you need to know about Integral Of 1 1 X 2 3 2 Solved With Clear Reasoning
Is this a definite integral?
No. The expression as written is most naturally an indefinite integral because no limits of integration appear.
Can this be solved without trig substitution?
For this form, trig substitution is the standard elegant method, though some alternative derivations can be built from reverse differentiation once the answer is known.
What if the intended integral was different?
If the original problem was meant to be $$\int \frac{1}{1+x^2}\,dx$$, the answer would instead be $$\arctan(x)+C$$. The spacing in the prompt strongly suggests the $$(1+x^2)^{3/2}$$ version, though, because that exact pattern appears in standard integral references.
