Questions: Evaluate the integral [ int x^3(x^4-3)^9 dx ]

Solution Steps

To evaluate the integral \(\int x^{3}\left(x^{4}-3\right)^{9} d x\), we can use the substitution method. Let \( u = x^4 - 3 \), then \( du = 4x^3 dx \). This substitution simplifies the integral, allowing us to integrate with respect to \( u \).

To evaluate the integral \(\int x^{3}\left(x^{4}-3\right)^{9} \, dx\), we use the substitution method. Let \( u = x^4 - 3 \), which implies \( du = 4x^3 \, dx \) or \( x^3 \, dx = \frac{1}{4} \, du \).

Substituting \( u \) into the integral, we have: \[ \int x^{3}\left(x^{4}-3\right)^{9} \, dx = \int \frac{1}{4} u^9 \, du = \frac{1}{4} \int u^9 \, du \]

Integrate \( \frac{1}{4} \int u^9 \, du \): \[ \frac{1}{4} \cdot \frac{u^{10}}{10} = \frac{1}{40} u^{10} \]

Substitute back \( u = x^4 - 3 \) into the expression: \[ \frac{1}{40} (x^4 - 3)^{10} \]

The expanded form of the integral, as calculated, is: \[ \frac{x^{40}}{40} - \frac{3x^{36}}{4} + \frac{81x^{32}}{8} - 81x^{28} + \frac{1701x^{24}}{4} - \frac{15309x^{20}}{10} + \frac{15309x^{16}}{4} - 6561x^{12} + \frac{59049x^{8}}{8} - \frac{19683x^{4}}{4} \]

Final Answer