Integral Calculator with Steps

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u-substitution integral calculator

u-substitution integral calculator, step by step

Substitution undoes the chain rule. When the integrand has the form f(g(x))·g′(x), replacing g(x) with u turns the integral into ∫ f(u) du. The tool tests the inner functions of the integrand as candidates, checks that the leftover x disappears and shows u, du and the way back.

How to find the u that works

The natural candidate is the inner function: the argument of a sine, the base of a power, the exponent of an exponential or the denominator. Compute du = g′(x) dx and check whether the rest of the integrand, divided by g′(x), depends only on u. In 2x·cos(x²), u = x² gives du = 2x dx, cos u is left, and the result is sin(x²) + C. In x·(x² + 1)⁵, u = x² + 1 gives du = 2x dx, so x dx = du/2 and the result is (x² + 1)⁶/12 + C.
Sometimes an x is left after the swap. If u is linear, as in u = x + 1, just write x = u − 1 and the integral becomes a sum of powers. That is why ∫ x·√(x + 1) dx closes at (2/5)(x + 1)^(5/2) − (2/3)(x + 1)^(3/2) + C. If u = ln x, then x = eᵘ. The tool applies these rewrites and shows the extra line, so the step does not look like magic.
On the way back, replace u with the original expression. Forgetting this is the most common mistake: the answer must be a function of x. In definite integrals there are two equivalent options, going back to x or converting the limits with u(a) and u(b); the tool goes back to x and evaluates F(b) − F(a), which avoids botching the limit conversion.

Examples solved by the tool

Inner function x²

Input: ∫ 2x·cos(x^2) dx
Expected output: sen(x²) + C

u = x², du = 2x dx.

Composite power

Input: ∫ x·(x^2+1)^5 dx
Expected output: (x² + 1)⁶/12 + C

u = x² + 1.

Exponential quotient

Input: ∫ e^x/(1+e^x) dx
Expected output: ln(1 + eˣ) + C

u = 1 + eˣ, du = eˣ dx.

Logarithm over x

Input: ∫ ln(x)/x dx
Expected output: (ln x)²/2 + C

u = ln x, du = dx/x.

Linear argument

Input: ∫ 1/(3x+2) dx
Expected output: ln|3x + 2|/3 + C

u = 3x + 2, du = 3 dx.

Full tool FAQ

What is the indefinite integral and why is there a constant C?

The indefinite integral of f is the family of functions F whose derivative is f. Since the derivative of a constant is zero, if F is an antiderivative then F + C is too, for any number C. That is why ∫ 2x dx = x² + C: x² + 1 and x² − 7 also have derivative 2x. The constant can only be pinned down with an extra condition, such as F(0) = 3.

How does the calculator check that the answer is right?

Which integration techniques does it show in the step by step?

How do I choose u in integration by parts?

How do you compute a definite integral?

What is an improper integral and how do I tell if it converges?

Why do some integrals have no answer, such as e^(−x²)?

How do I type the function? Does it accept sin, tan, roots and absolute value?

My notebook answer is different. Who is right?

What are the tool's limits? Does my data leave the browser?

Frequently asked questions

What if no u works?

Then the method may be another one: parts for products such as x·eˣ, partial fractions for quotients of polynomials, or trigonometric identities for powers of sin and cos. The calculator goes through all these rules on its own, in the order in which they usually solve fastest.

Do I need the absolute value in the logarithm?

Yes when the argument can be negative: ∫ du/u = ln|u| + C. The tool drops the absolute value when the argument is always positive, as in ln(1 + eˣ) or ln(x² + 1).

Must the result equal the answer key?

Not literally. Two antiderivatives differ by a constant, and the same function can be written in different forms. Differentiate the answer key: if you get the integrand, it is equivalent.

Next steps

x	f(x)	F′(x)	Difference
-2.26376	-0.23533751	-0.23533751	2.8e-17
0.565523	0.99552794	0.99552794	1.1e-16
4.44149	377.08727	377.08727	5.7e-14
-0.222645	-0.17820488	-0.17820488	0
-0.357476	-0.25003303	-0.25003303	0
0.460526	0.72989202	0.72989202	2.2e-16

f(x)

F′(x)

Difference

-2.26376

-0.23533751

2.8e-17

0.565523

0.99552794

1.1e-16

4.44149

377.08727

5.7e-14

-0.222645

-0.17820488

-0.357476

-0.25003303

0.460526

0.72989202

2.2e-16

How to find the u that works

The natural candidate is the inner function: the argument of a sine, the base of a power, the exponent of an exponential or the denominator. Compute du = g′(x) dx and check whether the rest of the integrand, divided by g′(x), depends only on u. In 2x·cos(x²), u = x² gives du = 2x dx, cos u is left, and the result is sin(x²) + C. In x·(x² + 1)⁵, u = x² + 1 gives du = 2x dx, so x dx = du/2 and the result is (x² + 1)⁶/12 + C.

Sometimes an x is left after the swap. If u is linear, as in u = x + 1, just write x = u − 1 and the integral becomes a sum of powers. That is why ∫ x·√(x + 1) dx closes at (2/5)(x + 1)^(5/2) − (2/3)(x + 1)^(3/2) + C. If u = ln x, then x = eᵘ. The tool applies these rewrites and shows the extra line, so the step does not look like magic.

On the way back, replace u with the original expression. Forgetting this is the most common mistake: the answer must be a function of x. In definite integrals there are two equivalent options, going back to x or converting the limits with u(a) and u(b); the tool goes back to x and evaluates F(b) − F(a), which avoids botching the limit conversion.

Examples solved by the tool

Inner function x²

Input: ∫ 2x·cos(x^2) dx
Expected output: sen(x²) + C

u = x², du = 2x dx.

Composite power

Input: ∫ x·(x^2+1)^5 dx
Expected output: (x² + 1)⁶/12 + C

u = x² + 1.

Exponential quotient

Input: ∫ e^x/(1+e^x) dx
Expected output: ln(1 + eˣ) + C

u = 1 + eˣ, du = eˣ dx.

Logarithm over x

Input: ∫ ln(x)/x dx
Expected output: (ln x)²/2 + C

u = ln x, du = dx/x.

Linear argument

Input: ∫ 1/(3x+2) dx
Expected output: ln|3x + 2|/3 + C

u = 3x + 2, du = 3 dx.

Full tool FAQ

What is the indefinite integral and why is there a constant C?

How does the calculator check that the answer is right?

Which integration techniques does it show in the step by step?

How do I choose u in integration by parts?

How do you compute a definite integral?

What is an improper integral and how do I tell if it converges?

Why do some integrals have no answer, such as e^(−x²)?

How do I type the function? Does it accept sin, tan, roots and absolute value?

My notebook answer is different. Who is right?

What are the tool's limits? Does my data leave the browser?

Frequently asked questions

What if no u works?

Do I need the absolute value in the logarithm?

Yes when the argument can be negative: ∫ du/u = ln|u| + C. The tool drops the absolute value when the argument is always positive, as in ln(1 + eˣ) or ln(x² + 1).

Must the result equal the answer key?

Not literally. Two antiderivatives differ by a constant, and the same function can be written in different forms. Differentiate the answer key: if you get the integrand, it is equivalent.

Integral Calculator with Steps

Continue in the tool

More pages for this tool

u-substitution integral calculator, step by step

How to find the u that works

Examples solved by the tool

Inner function x²

Composite power

Exponential quotient

Logarithm over x

Linear argument

Full tool FAQ

Frequently asked questions

What if no u works?

Do I need the absolute value in the logarithm?

Must the result equal the answer key?

Next steps

Integral Calculator with Steps

Result

Step by step

Graph

Continue in the tool

More pages for this tool

u-substitution integral calculator, step by step

How to find the u that works

Examples solved by the tool

Inner function x²

Composite power

Exponential quotient

Logarithm over x

Linear argument

Full tool FAQ

Frequently asked questions

What if no u works?

Do I need the absolute value in the logarithm?

Must the result equal the answer key?

Next steps

Result

Step by step

Graph