Solve any maths problem

Equations, derivatives, integrals, matrices, triangles, primes, statistics, or a word problem the tutor breaks into parts.

Calculus · Definite-Integrals

Integrate e^(-4t) from 0 to oo

\int_{0}^{\infty} e^{- 4 t}\, dt

Step by step

  1. \int_{0}^{\infty} e^{- 4 t}\, dt

    First find an antiderivative F, then evaluate F(b) − F(a).

  2. u = - 4 t,\quad du = -4\, dt

    Substitute u = - 4 t.

  3. \int e^{- 4 t}\, dt = \int - \frac{e^{u}}{4}\, d_u

    Rewrite the integral in terms of u.

  4. \int - \frac{e^{u}}{4}\, d_u = - \frac{1}{4} \int e^{u}\, d_u

    Pull the constant - \frac{1}{4} out of the integral.

  5. \int e^{u}\, d_u = e^{u}

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

  6. = - \frac{e^{- 4 t}}{4}

    Substitute back u = - 4 t.

  7. F(\infty) - F(0) = \left(0\right) - \left(- \frac{1}{4}\right)

    Fundamental theorem of calculus: plug in the limits.

  8. = \frac{1}{4}

    Simplify.

Reveal the answer
\frac{1}{4}