Question

\[\int\limits_0^1 \left( 3 x^2 + 5x \right) dx\]

Answer 1

\[\int_a^b f\left( x \right) d x = \lim_{h \to 0} h\left[ f\left( a \right) + f\left( a + h \right) + f\left( a + 2h \right) . . . . . . . . . . . . . . . + f\left( a + \left( n - 1 \right)h \right) \right]\]
\[\text{where }h = \frac{b - a}{n}\]

\[\text{Here }a = 0, b = 1, f\left( x \right) = 3 x^2 + 5x, h = \frac{1 - 0}{n} = \frac{1}{n}\]
Therefore,
\[I = \int_0^1 \left( 3 x^2 + 5x \right) d x\]
\[ = \lim_{h \to 0} h\left[ f\left( 0 \right) + f\left( 0 + h \right) + . . . . . . . . . . . . . . . . . . . . + f\left\{ 0 + \left( n - 1 \right)h \right\} \right]\]
\[ = \lim_{h \to 0} h\left[ \left( 0 + 0 \right) + \left( 3 h^2 + 5h \right) + . . . . . . . . . . . . . . . + \left\{ 3 \left( n - 1 \right)^2 h^2 + 5\left( n - 1 \right)h \right\} \right]\]
\[ = \lim_{h \to 0} h\left[ 5h\left( 1 + 2 + . . . . . . . . . + n \right) + 3 h^2 \left\{ 1^2 + 2^2 + 3^2 . . . . . . . . . + \left( n - 1 \right)^2 \right\} \right]\]
\[ = \lim_{h \to 0} h\left[ 5h\frac{n\left( n - 1 \right)}{2} + h^2 \frac{3n\left( n - 1 \right)\left( 2n - 1 \right)}{6} \right]\]
\[ = \lim_{n \to \infty} \frac{1}{n}\left[ \frac{5\left( n - 1 \right)}{2} + \frac{\left( n - 1 \right)\left( 2n - 1 \right)}{2n} \right]\]
\[ = \lim_{n \to \infty} \left[ \frac{5}{2}\left( 1 - \frac{1}{n} \right) + \frac{1}{2}\left( 1 - \frac{1}{n} \right)\left( 2 - \frac{1}{n} \right) \right]\]
\[ = \frac{5}{2} + 1\]
\[ = \frac{7}{2}\]