(D) Given the definite integral: $\int_{1}^{k}(3x^{2}+2x+1)dx=11$ Integrating the function term by term: $[x^{3}+x^{2}+x]_{1}^{k}=11$ Applying the lim

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(D) Given the definite integral: $\int_{1}^{k}(3x^{2}+2x+1)dx=11$ Integrating the function term by term: $[x^{3}+x^{2}+x]_{1}^{k}=11$ Applying the limits: $(k^{3}+k^{2}+k)-(1^{3}+1^{2}+1)=11$ Simplifying the expression: $k^{3}+k^{2}+k-3=11$ $k^{3}+k^{2}+k-14=0$ Testing for roots,if $k=2$: $(2)^{3}+(2)^{2}+2-14 = 8+4+2-14 = 0$ Since $k=2$ satisfies the equation,the value of $k$ is $2$.

Class 12 Mathematics 7-2.Definite Integral Fundamental definite integration

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If $\int_{1}^{k}(3x^{2}+2x+1)dx=11$,then $k=$

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$1/2$
B
$-2$
C
$-1/2$
D
$2$

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