maths.free › Calculus › Integrals

Integrals

Antiderivatives by substitution, parts and partial fractions.

Integration undoes differentiation. The solver shows which technique it uses — the power rule, a u-substitution, integration by parts, partial fractions — and the graph plots the antiderivative F against the integrand f, so you can see that F's slope is f.

Mfano wenye matokeo: integral of x*e^x dx

Calculus · Integrals

Integrate x·e^(x)

\int x e^{x}\, dx

Hatua kwa hatua

  1. \int x e^{x}\, dx

    Find an antiderivative.

  2. u = x,\quad dv = e^{x}\, dx

    Integration by parts: ∫u dv = uv − ∫v du.

  3. du = 1\, dx,\quad v = e^{x}

    Differentiate u, integrate dv.

  4. \int x e^{x}\, dx = x e^{x} - \int e^{x}\, dx

    Apply the formula.

  5. \int e^{x}\, dx = e^{x}

    ∫ aᵘ du = aᵘ / ln a (for eˣ that is just eˣ).

  6. = x e^{x} - e^{x}

    Combine.

  7. F(x) = \left(x - 1\right) e^{x} + C

    Add the constant of integration.

Lafunua jibu
\left(x - 1\right) e^{x} + C

Symbols used here

\int f(x)\,dx,\ \int_a^b
integral
Antiderivative (indefinite) or signed area from a to b (definite).
e
Euler's number
2.71828…, the base whose exponential is its own derivative.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
\pi
pi
Ratio of a circle's circumference to its diameter, 3.14159…
\infty
infinity
Not a number: "grows without bound" in limits and intervals.
\sin,\ \cos,\ \tan
sine, cosine, tangent
Ratios of sides in a right triangle; coordinates on the unit circle.
\log_b x,\ \ln x
logarithm, natural log
The exponent b must be raised to for x; ln uses base e.
\sum_{k=1}^{n} a_k
summation
Add a_k for k = 1 up to n.
\lim_{x \to a} f(x)
limit
The value f(x) approaches as x approaches a.
f'(x),\ \frac{dy}{dx}
derivative
Instantaneous rate of change; slope of the graph.
y',\ y''
first and second derivative of y
Prime notation for derivatives with respect to x (or t).
C,\ C_1,\ C_2
arbitrary constants
Constants of integration fixed by initial conditions.

How to: Integrals

  1. Find an antiderivative.
  2. Integration by parts: ∫u dv = uv − ∫v du.
  3. Differentiate u, integrate dv.
  4. Apply the formula.
  5. ∫ aᵘ du = aᵘ / ln a (for eˣ that is just eˣ).
  6. Combine.
  7. Add the constant of integration.

Questions people ask

What is a derivative in one sentence?

The slope of the graph at a point — the rate at which the output is changing there. Speed is the derivative of position.

What is an integral in one sentence?

The accumulated total of a rate — the area under the curve. Distance travelled is the integral of speed.

Why are derivatives and integrals opposites?

That is the fundamental theorem of calculus: accumulating a rate and then measuring how fast the accumulation grows gets you back the rate. Integration undoes differentiation up to a constant.

When do I use substitution and when integration by parts?

Substitution when part of the integrand is the derivative of another part (u and du both present). Parts when the integrand is a product of two unrelated kinds of function — a polynomial times an exponential, log or trig function.

Jaribu kufanya mambo yako mwenyewe

Mengi zaidi katika Calculus

LimitsDerivativesDefinite integralsTaylor seriesSeries and sumsMaxima and minimaThe chain ruleImplicit differentiationRelated rates and optimisationIntegration techniques: substitution, parts, partial fractionsApplications of integration: area, volume, arc lengthInfinite series and convergence tests