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This u-substitution calculator solves integrals by substitution and shows exactly how. It tells you what to pick for \(u\), finds \(du\), rewrites the whole integral in terms of \(u\), solves it, and then puts \(x\) back in. Every answer is checked by differentiation.
How to Use It
- Type the integrand, for example
2x(x^2+1)^5orx cos(x^2). - Press Solve.
- Read the U-Substitution step. It shows \(u = \ldots\) and \(du = \ldots\,dx\), then the new integral in \(u\).
If an integral does not need substitution (like \(\int x^2\,dx\)), the calculator still solves it and shows a note.
What Is U-Substitution?
U-substitution is the chain rule run backwards. If your integral has a function inside another function, and the derivative of the inside is also there, you can rename the inside as \(u\):
\[\int f(g(x))\,g'(x)\,dx = \int f(u)\,du, \quad u = g(x).\]
The hard integral turns into an easy one. For a full lesson with 7 examples, read our u-substitution guide.
How to Choose u
Look for the “inside” part whose derivative also shows up. Good choices are often:
- The expression inside parentheses raised to a power: \((x^2+1)^5\) → \(u = x^2+1\).
- The inside of a function: \(\cos(x^2)\) → \(u = x^2\).
- An exponent: \(e^{\sin x}\) → \(u = \sin x\).
- A denominator: \(\frac{2x}{x^2+9}\) → \(u = x^2+9\).
- A log: \(\frac{\ln x}{x}\) → \(u = \ln x\).
If the derivative is only off by a number, like 2 or 3, that is fine. You fix it with a constant.
Worked Examples
Example 1: \(\int 2x(x^2+1)^5\,dx\)
Let \(u = x^2 + 1\). Then \(du = 2x\,dx\). The integral becomes \(\int u^5\,du = \frac{u^6}{6} + C\).
Answer: \(\frac{(x^2+1)^6}{6} + C\).
Example 2: \(\int x\cos(x^2)\,dx\)
Let \(u = x^2\), so \(du = 2x\,dx\) and \(x\,dx = \frac{du}{2}\). Then \(\frac12\int\cos u\,du = \frac12\sin u + C\).
Answer: \(\frac12\sin(x^2) + C\).
Example 3: \(\int e^{\sin x}\cos x\,dx\)
Let \(u = \sin x\), so \(du = \cos x\,dx\). Then \(\int e^u\,du = e^u + C\).
Answer: \(e^{\sin x} + C\).
Example 4: \(\int \frac{1}{x\ln x}\,dx\)
Let \(u = \ln x\), so \(du = \frac{1}{x}\,dx\). Then \(\int\frac1u\,du = \ln|u| + C\).
Answer: \(\ln|\ln x| + C\).
Example 5: \(\int x\sqrt{x+1}\,dx\) (back-substitution)
Let \(u = x + 1\). There is an extra \(x\), so write \(x = u – 1\). Then \(\int (u-1)u^{1/2}\,du = \int (u^{3/2} – u^{1/2})\,du = \frac25u^{5/2} – \frac23u^{3/2} + C\).
Substitution in Definite Integrals
When the integral has limits, change the limits to \(u\)-values too. For \(\int_0^1 2x(x^2+1)^3\,dx\), the limits become \(u = 1\) and \(u = 2\). Then you never have to switch back to \(x\). Try it in the definite integral calculator.
Common Mistakes
- Leaving some x behind. After substituting, only \(u\) and \(du\) should remain.
- Forgetting to convert dx. \(dx\) is not the same as \(du\).
- Not substituting back. The final answer must be in terms of \(x\).
- Using substitution for a product that needs parts. \(\int x e^x\,dx\) has no good \(u\). Use the integration by parts calculator.
Learn the Math Behind the U-Substitution Calculator
U-substitution is the reverse chain rule. It works when the derivative of an inside function is also in the integral.
Frequently Asked Questions
How does the calculator pick u?
It lists the “inside” parts of your function, from largest to smallest. For each one, it checks if dividing by \(du\) leaves only \(u\). The first choice that works is shown to you.
What if the badge says “Solved without u-substitution”?
That means a simpler rule worked, or a different method was needed. The steps still show the full solution.
Is u-substitution the same as change of variables?
Yes. Some books call it integration by substitution or change of variables.
Related Guides
- U-substitution: step-by-step guide with 7 worked examples
- Top 5 mistakes students make with antiderivatives
- Antiderivative practice problems