Free Integral Calculator with Steps
Calculate integrals step-by-step. Learn power rule, substitution, integration by parts, and the fundamental theorem of calculus. Visualise the antiderivative curve and shaded area on an interactive graph. Perfect for A-Level and university calculus.
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Enter a mathematical expression above to learn how to integrate it step-by-step.
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What is an Integral?
An integral is the reverse process of differentiation. The indefinite integral (antiderivative) finds a function whose derivative equals the original function. The definite integral calculates the signed area under a curve between two points.
The Fundamental Theorem of Calculus connects these concepts: the definite integral equals F(b) − F(a), where F is any antiderivative of the integrand. This powerful theorem means we can evaluate areas by finding antiderivatives rather than summing infinitesimal rectangles.
Real-world applications: Integrals calculate total distance from velocity, work done by a variable force, area and volume of shapes, centre of mass, probability distributions, and accumulated quantities across all sciences.
Basic Integration Rules
Common Integrals
Trigonometric Functions
∫sin(x) dx = -cos(x) + C
∫cos(x) dx = sin(x) + C
∫sec²(x) dx = tan(x) + C
∫csc²(x) dx = -cot(x) + C
Exponential & Logarithmic
∫eˣ dx = eˣ + C
∫aˣ dx = aˣ/ln(a) + C
∫1/x dx = ln|x| + C
Inverse Trigonometric
∫1/√(1-x²) dx = arcsin(x) + C
∫1/(1+x²) dx = arctan(x) + C
Indefinite vs Definite
Indefinite Integral
\u222Bf(x) dx = F(x) + C — Finds the antiderivative, a family of functions differing by a constant.
Definite Integral
\u222B[a,b] f(x) dx = F(b) − F(a) — Calculates the signed area under the curve from x=a to x=b.
Integration Techniques & Methods
Power Rule
∫xⁿ dx = xⁿ⁺¹/(n+1) + C
Add 1 to the exponent and divide by the new exponent. Works for all real n ≠ -1.
e.g. ∫x³ dx = x⁴/4 + C
u-Substitution
∫f(g(x))·g′(x) dx = ∫f(u) du
The reverse chain rule. Identify the inner function u = g(x), substitute, integrate, then back-substitute.
e.g. ∫2x·cos(x²) dx = sin(x²) + C
Integration by Parts
∫u dv = uv - ∫v du
For products of functions. Choose u using LIATE: Logarithmic, Inverse trig, Algebraic, Trig, Exponential.
e.g. ∫x·eˣ dx = xeˣ - eˣ + C
Trig Integrals
sin→-cos, cos→sin, sec²→tan
Standard integrals for trig functions. For powers like sin²(x), use double angle identities.
e.g. ∫sin²(x) dx = x/2 - sin(2x)/4 + C
Partial Fractions
Split rational functions into simpler fractions
Decompose P(x)/Q(x) into partial fractions, then integrate each term separately.
e.g. ∫1/(x²-1) dx = ½ ln|x-1| - ½ ln|x+1| + C
Fundamental Theorem
∫[a,b] f(x) dx = F(b) - F(a)
Connects antiderivatives to definite integrals. Find F(x), then evaluate at bounds.
e.g. ∫[0,π] sin(x) dx = [-cos(x)]₀ᵘ = 2
Common Integration Mistakes
Forgetting + C on indefinite integrals
∫x² dx = x³/3
∫x² dx = x³/3 + C
Every indefinite integral must include the constant of integration. Examiners always check for this.
Wrong power rule formula (dividing by n)
∫x² dx = x²/2 + C
∫x² dx = x³/3 + C
Add 1 to the exponent FIRST, then divide by the new exponent (n+1), not the old one (n).
Wrong sign on trig integrals
∫sin(x) dx = cos(x) + C
∫sin(x) dx = -cos(x) + C
Remember: integrating sin gives NEGATIVE cos. Check by differentiating your answer.
Forgetting chain rule factor in substitution
∫cos(3x) dx = sin(3x) + C
∫cos(3x) dx = sin(3x)/3 + C
When the argument is kx, divide by k. This is the "reverse chain rule" adjustment.
Not back-substituting after u-sub
Answer left as F(u) instead of F(g(x))
Always replace u with the original expression g(x)
Your final answer must be in terms of the original variable, never in terms of u.
Confusing ∫eˣ dx with ∫xᵉ dx
∫xᵉ dx = xᵉ (treating as exponential)
∫xᵉ dx = xᵉ⁺¹/(e+1) + C (power rule)
Variable in base = power rule. Variable in exponent = exponential rule.
Worked Examples
∫(3x² - 4x + 7) dx
f(x) = 3x² - 4x + 7
Integrate term by term:
∫3x² dx = 3 × x³/3 = x³
∫-4x dx = -4 × x²/2 = -2x²
∫7 dx = 7x
F(x) = x³ - 2x² + 7x + C
∫2x·cos(x²) dx
Let u = x², then du = 2x dx
Substitute: ∫cos(u) du
Integrate: sin(u) + C
Back-substitute: sin(x²) + C
∫2x·cos(x²) dx = sin(x²) + C
∫x·eˣ dx
Choose: u = x, dv = eˣ dx
Then: du = dx, v = eˣ
Apply ∫u dv = uv - ∫v du:
= x·eˣ - ∫eˣ dx
= x·eˣ - eˣ + C
= eˣ(x - 1) + C
∫[0,π] sin(x) dx
∫sin(x) dx = -cos(x)
Evaluate at bounds:
F(π) = -cos(π) = -(-1) = 1
F(0) = -cos(0) = -(1) = -1
F(π) - F(0) = 1 - (-1) = 2
∫[0,π] sin(x) dx = 2
Exam Tips for Integration
Always add + C to indefinite integrals
Forgetting the constant of integration is the most common mark lost in integration questions. Make it a habit to write + C immediately after integrating.
Show substitution working clearly
Write "Let u = ..., du = ... dx" explicitly. Examiners award method marks for showing the substitution process, even if you make an error later.
For parts: choose u wisely (LIATE rule)
Follow LIATE priority: Logarithmic, Inverse trig, Algebraic, Trig, Exponential. Choose u as the function earlier in this list for easier integration.
Check by differentiating your answer
The quickest way to verify an integral is to differentiate your answer. If you get back the original function, your integral is correct.
For definite integrals, evaluate bounds carefully
Use square bracket notation [F(x)]_a^b and substitute systematically. Common errors: wrong sign, evaluating at wrong bound, or forgetting to subtract F(a).
Simplify before integrating when possible
Expand brackets, simplify fractions, or use identities before integrating. For example, ∫(x+1)² dx is easier if you expand to x² + 2x + 1 first.
Common Integrals Reference Table
Basic Functions
Trigonometric
Exp, Log & Inverse Trig
With the reverse chain rule: \u222Bsin(kx) dx = -cos(kx)/k + C, \u222Be^(kx) dx = e^(kx)/k + C
Frequently Asked Questions
How do I integrate a function?
Identify the function type and apply the reverse of differentiation rules. Use power rule for polynomials, standard integrals for trig/exp/log, or advanced techniques like substitution and parts for composite or product functions.
What's the difference between definite and indefinite integrals?
Indefinite gives antiderivative F(x) + C (a family of functions). Definite has bounds [a,b] and gives a specific number: the signed area under the curve, calculated as F(b) - F(a).
When do I use substitution vs integration by parts?
Substitution for composite functions (reverse chain rule). Parts for products of functions using ∫u dv = uv - ∫v du. If you see f(g(x))·g′(x), use substitution. If you see a product like x·eˣ, use parts.
Why do we add "+ C"?
Any constant disappears when differentiated. The + C represents all possible constants that give the same derivative. Without it, you only have one specific antiderivative rather than the general solution.
How do I find area under a curve?
Use definite integral: ∫[a,b] f(x) dx = F(b) - F(a). This gives signed area between curve and x-axis. For area between curves, integrate |f(x) - g(x)|.
Is this suitable for A-Level revision?
Yes! Covers power rule, trig integrals, exponentials, substitution, and parts — all A-Level techniques. The Learn Mode teaches the exact methods examiners expect, with step-by-step working.
What is the Fundamental Theorem of Calculus?
It connects differentiation and integration: if F is an antiderivative of f, then ∫[a,b] f(x) dx = F(b) - F(a). This means you evaluate definite integrals by finding antiderivatives.
What is integration by parts?
A technique for integrating products: ∫u dv = uv - ∫v du. Use the LIATE rule to choose u (pick the function type earliest in: Log, Inverse trig, Algebraic, Trig, Exponential).
How do I integrate trigonometric functions?
Standard: ∫sin(x) = -cos(x), ∫cos(x) = sin(x), ∫sec²(x) = tan(x). For powers like sin²(x), use the identity: sin²(x) = (1 - cos(2x))/2 before integrating.
What is u-substitution?
The reverse of the chain rule. Identify u = g(x), compute du = g′(x) dx, rewrite the integral in terms of u, integrate, then substitute back. Essential for composite functions.
Can this calculator handle definite integrals?
Yes! Toggle the Definite switch, enter your bounds, and click Integrate. The calculator finds the antiderivative, evaluates at both bounds, and shows the shaded area on the graph.
How do I find the average value of a function?
Average value on [a,b] = (1/(b-a)) × ∫[a,b] f(x) dx. Compute the definite integral first, then divide by the interval width (b-a).
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