Mathematics

Let $$F(x)$$ be the primitive of $$\displaystyle \frac{3x+2}{\sqrt{x-9}}$$ w.r.t $$x$$ . If $$F(10)=60$$ then the value of $$F(13)$$, is


ANSWER

$$132$$


SOLUTION

$$F(x)=\displaystyle \int \frac{3x+2}{\sqrt{x-9}}dx$$
Let, $$\displaystyle \sqrt{x-9}=t \Rightarrow  \frac{1}{2\sqrt{x-9}}=\frac{\mathrm{d} t}{\mathrm{d} x}$$
$$x-9=t^{2} \Rightarrow  x=t^{2}+9$$
So, $$\displaystyle \int \frac{3x+2}{\sqrt{x-9}}dx=\int 2(3t^{2}+29)dt=2(t^{3}+29t)+c$$
$$F(10)=60, t=1,$$
$$2(1+29)+c=60 \Rightarrow  c=0$$
$$F(13) \Rightarrow  t=2,$$
$$=2(8+58)=2(66)=132$$

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