Answer :
The solution of the given integral is [tex]\[ \int_0^1 7x^6 e^{x^7} \, dx = e - 1 \][/tex].
To evaluate the integral [tex]\( \int_0^1 7x^6 e^{x^7} \, dx \)[/tex], we can make the substitution [tex]\( u = x^7 \)[/tex].
Then, [tex]\( du = 7x^6 \, dx \)[/tex], and [tex]\( dx = \frac{1}{7x^6} \, du \).[/tex]
Now, let's rewrite the integral using the substitution:
[tex]\[ \int_0^1 7x^6 e^{x^7} \, dx = \int_{u(0)}^{u(1)} e^u \, du \][/tex]
Now, we need to find the limits of integration in terms of u:
[tex]\[ u(0) = (0)^7 = 0 \][/tex]
[tex]\[ u(1) = (1)^7 = 1 \][/tex]
So, the integral becomes:
[tex]\[ \int_0^1 e^u \, du \][/tex]
The antiderivative of [tex]\( e^u \)[/tex] is [tex]\( e^u \)[/tex], so the integral evaluates to:
[tex]\[ \int_0^1 e^u \, du = \left[ e^u \right]_0^1 = e^1 - e^0 = e - 1 \][/tex]
Therefore,
[tex]\[ \int_0^1 7x^6 e^{x^7} \, dx = e - 1 \][/tex]
