Find the antiderivative $F (x)$ of the function $f (x) = 3 x^{4} - 2 x + 5$ . Then compute the definite integral $\int_{0}^{2} f (x) d x$ .

easy 2 marks Class 12 · Mathematics · Integrals

Answer

\frac{126}{5}

The correct answer is $\frac{126}{5}$ — Therefore $\int_{0}^{2} f (x) d x = \frac{126}{5}$ .

Step-by-step solution

Write $f (x) = 3 x^{4} - 2 x + 5$ .
Integrate term by term: $\int 3 x^{4} d x = \frac{3}{5} x^{5}$ , $\int (- 2 x) d x = - x^{2}$ , $\int 5 d x = 5 x$ .
Hence the general antiderivative is $F (x) = \frac{3}{5} x^{5} - x^{2} + 5 x + C$ .
For a definite integral the constant $C$ cancels, so evaluate $F (2) - F (0)$ .
Compute $F (2) = \frac{3}{5} (2)^{5} - (2)^{2} + 5 (2) = \frac{3}{5} \cdot 32 - 4 + 10 = \frac{96}{5} + 6 = \frac{126}{5}$ .
Compute $F (0) = 0$ .
Therefore $\int_{0}^{2} f (x) d x = \frac{126}{5}$ .

Common mistake: Missing the

\frac{1}{5}

factor when integrating

x^{4}

or forgetting that the constant

C

cancels in a definite integral.

Concept tested: Antiderivative and definite integral