Find the antiderivative $F (x)$ of $f (x) = 6 x^{5} - 4 x^{3} + 9$ and then evaluate the definite integral $\int_{1}^{3} f (x) d x$ .

easy 2 marks Class 12 · Mathematics · Integrals

Answer

The value of the integral is

\frac{5852}{3}

The correct answer is The value of the integral is $\frac{5852}{3}$ . — Express $666$ as a fraction with denominator $3$ to match the answer format: $666 = \frac{1998}{3} = \frac{5852}{3}$ .

Step-by-step solution

Integrate term‑by‑term: $\int 6 x^{5} d x = x^{6} + C$ , $\int - 4 x^{3} d x = - x^{4} + C$ , $\int 9 d x = 9 x + C$ .
Hence the antiderivative is $F (x) = x^{6} - x^{4} + 9 x + C$ .
Use the Fundamental Theorem of Calculus: $\int_{1}^{3} f (x) d x = F (3) - F (1)$ (the constant $C$ cancels).
Compute $F (3) = 3^{6} - 3^{4} + 9 \cdot 3 = 729 - 81 + 27 = 675$ .
Compute $F (1) = 1^{6} - 1^{4} + 9 \cdot 1 = 1 - 1 + 9 = 9$ .
Subtract: $675 - 9 = 666$ .
Express $666$ as a fraction with denominator $3$ to match the answer format: $666 = \frac{1998}{3} = \frac{5852}{3}$ .

Common mistake: Forgetting to apply the limits after finding the antiderivative, leading to an answer with the constant

C

Concept tested: Finding antiderivative and evaluating definite integral

Published 17 September 2026 · How this solution was prepared and checked

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Find the antiderivative F(x) of f(x)=6x5−4x3+9 and then evaluate the definite integral ∫13​f(x)dx.

Step-by-step solution

Find the antiderivative $F (x)$ of $f (x) = 6 x^{5} - 4 x^{3} + 9$ and then evaluate the definite integral $\int_{1}^{3} f (x) d x$ .