Question

evaluate the integral \\( \int 28t^6 e^{-t^7} \\, dt \\).
\\( \int 28t^6 e^{-t^7} \\, dt = \square \\)

Explanation:

Step1: Choose substitution

Let \( u = -t^7 \), then find \( du \).
Differentiate \( u \) with respect to \( t \): \( \frac{du}{dt} = -7t^6 \), so \( du = -7t^6 dt \), and \( -4du = 28t^6 dt \).

Step2: Substitute into integral

The integral \( \int 28t^6 e^{-t^7} dt \) becomes \( \int e^{u} \cdot (-4du) \) (wait, correction: from \( du = -7t^6 dt \), we can see that \( 28t^6 dt = -4 du \)? No, actually, \( 28t^6 dt = -4 \times (-7t^6 dt)= -4 du \)? Wait, no: \( du = -7t^6 dt \implies 28t^6 dt = -4 du \)? Wait, \( 28t^6 dt = 28 \times t^6 dt \), and \( du = -7t^6 dt \implies t^6 dt = -\frac{du}{7} \), so \( 28t^6 dt = 28 \times (-\frac{du}{7}) = -4 du \). Then the integral is \( \int e^{u} \times (-4 du) \)? Wait, no, wait: the original integrand is \( 28t^6 e^{-t^7} dt \), and \( u = -t^7 \), so \( e^{-t^7}=e^u \), and \( 28t^6 dt \): since \( du = -7t^6 dt \), then \( 28t^6 dt = -4 du \) (because \( 28 / (-7) = -4 \)). So the integral becomes \( \int e^u \times (-4 du) \)? Wait, no, that's a mistake. Wait, \( du = -7t^6 dt \implies t^6 dt = -\frac{du}{7} \), so \( 28t^6 dt = 28 \times (-\frac{du}{7}) = -4 du \). So the integral is \( \int e^u \times (-4 du) \)? Wait, no, the integrand is \( 28t^6 e^{-t^7} dt = 28t^6 dt \times e^{-t^7} \), and \( e^{-t^7}=e^u \), \( 28t^6 dt = -4 du \), so the integral is \( \int e^u \times (-4 du) \)? Wait, no, that would be \( -4 \int e^u du \), but let's check again. Wait, maybe I messed up the sign. Let's do it again:

Let \( u = -t^7 \). Then \( du/dt = -7t^6 \implies dt = du / (-7t^6) \). Substitute into the integral:

\( \int 28t^6 e^{u} \times (du / (-7t^6)) \). The \( t^6 \) cancels, 28 / (-7) = -4, so we have \( \int -4 e^u du = -4 \int e^u du \).

Step3: Integrate \( e^u \)

The integral of \( e^u \) with respect to \( u \) is \( e^u + C \), so \( -4 \int e^u du = -4 e^u + C \).

Step4: Substitute back \( u \)

Since \( u = -t^7 \), substitute back: \( -4 e^{-t^7} + C \).

Wait, but let's check by differentiating the result. The derivative of \( -4 e^{-t^7} \) with respect to \( t \) is \( -4 \times e^{-t^7} \times (-7t^6) = 28t^6 e^{-t^7} \), which matches the integrand. So that's correct.

Answer:

\( -4e^{-t^7} + C \) (where \( C \) is the constant of integration)