Free Step-by-Step Integral Solver

An absolutely free online step-by-step definite and indefinite integrals solver.

Examples: $\int x^2 e^x\,dx$ $\int x \cos(x^2)\,dx$ $\int \frac{dx}{x^2-1}$ $\int \sin^3(x)\,dx$ $\int \sqrt{1-x^2}\,dx$ $\int\limits_0^1 x^2\,dx$ $\int\limits_0^\infty \frac{dx}{1+x^2}$

How to enter a function

• * is multiplication; it can be left out: 2x, x sin(x), (x+1)(x-2).
• x^2 is $x^2$, x^(1/3) is $x^{\frac13}$ and e^(2x) is $e^{2x}$.
• sqrt(x) is $\sqrt{x}$ and sqrt[3](x) is $\sqrt[3]{x}$.
• (a+b)/(c+d) is $\frac{a+b}{c+d}$. Put the numerator and the denominator in brackets.
• ln(x) or log(x) is the natural logarithm.
• sin(x), cos(x), tan(x), cot(x), sec(x), csc(x); sin^2(x) is $\sin^2(x)$.
• arcsin(x), arccos(x), arctan(x); sinh(x), cosh(x), tanh(x).
• pi is $\pi$, e is the number $e$ and oo is $\infty$ (only as a limit of a definite integral).
• The variable of integration is $x$. If there is no $x$, another letter is taken, or the one you end with: sin(t) dt. The other letters are constants.

Methods of integration

The solver finds the integral the way it is done by hand and shows every step. Click an example to see the method at work.

• The table of integrals and the linearity rules: $\int (3x^2+2x-5)\,dx$
• Substitution: $\int \frac{x}{x^2+1}\,dx$, $\int e^{\sqrt{x}}\,dx$
• Integration by parts: $\int x \ln(x)\,dx$, $\int e^x \sin(x)\,dx$
• Powers and products of trigonometric functions: $\int \sin^2(x) \cos^3(x)\,dx$, $\int \sin(2x) \cos(3x)\,dx$
• Rational functions (division, partial fractions, completing the square): $\int \frac{x^3+1}{x^2-x}\,dx$, $\int \frac{dx}{x^2+2x+5}$
• Trigonometric substitution: $\int \frac{x^2}{\sqrt{4-x^2}}\,dx$
• Definite and improper integrals by the Newton-Leibniz formula: $\int\limits_0^{\pi} x \sin(x)\,dx$, $\int\limits_0^{\infty} e^{-x}\,dx$

Every answer is checked: the derivative of the result must give the function under the integral. Some integrals, like $\int e^{-x^2}dx$ or $\int \frac{\sin x}{x}dx$, are not elementary functions; the solver writes them with the special functions $\text{erf}$, $\text{Si}$, $\text{Ci}$, $\text{Ei}$, $\text{li}$ and the Fresnel integrals. When no formula is found, a definite integral is still computed numerically.

Common Integrals

$\int 0dx = \text{const}$

$\int\ dx = x + \text{const}$

$\int kdx = kx + \text{const}$

$\int x^n\ dx = \frac{1}{n+1}x^{n+1} + \text{const}$ here n≠-1

$\int \frac{1}{x}\ dx = \int x^{-1}\ dx = \ln|x| + \text{const}$

$\int x^{-n}\ dx = \frac{1}{-n+1}x^{-n+1} + \text{const}$

$\int \frac{1}{ax+b}\ dx = \frac{1}{a}\ln|ax+b| + \text{const}$

$\int e^x\ dx = e^x + \text{const}$

$\int a^x\ dx = \frac{a^x}{\ln a} + \text{const}$

$\int \sin(x)\ dx = -\cos(x) + \text{const}$

$\int \cos(x)\ dx = \sin(x) + \text{const}$

$\int \tan(x)\ dx = \ln|\sec(x)| + \text{const}$

$\int \cot(x)\ dx = \ln|\sin(x)| + \text{const}$

$\int \frac{1}{\sqrt{1-x^2}} \ dx = \arcsin(x) + \text{const}$

$\int -\frac{1}{\sqrt{1-x^2}} \ dx = \arccos(x) + \text{const}$

$\int \frac{1}{1+ x^2}\ dx = \arctan(x) + \text{const}$

$\int -\frac{1}{1+x^2}\ dx = \text{arccot}(x) + \text{const}$

Integration by Parts

$\int u\ dv = uv - \int v\ du$

$\int\limits_{a}^{b} u\ dv = uv |_a^b - \int\limits_{a}^{b} v\ du$

Trigonometric Substitutions

$\sqrt{a^2 - b^2x^2}$ $\Rightarrow x=\frac{a}{b}\sin\theta$ and $\cos^2\theta = 1 - \sin^2\theta$

$\sqrt{a^2 + b^2x^2}$ $\Rightarrow x=\frac{a}{b}\tan\theta$ and $\sec^2\theta = 1 + \tan^2\theta$

$\sqrt{b^2x^2 - a^2}$ $\Rightarrow x=\frac{a}{b}\sec\theta$ and $\tan^2\theta = \sec^2\theta - 1$