Fundamental Theorem of Calculus

Rarely Tested

$$\int_a^b f(x) \, dx = F(b) - F(a)$$
where $$F'(x) = f(x)$$. Evaluates definite integral using antiderivative.

Question 1

The value of the definite integral

$$\int_{0}^{2}\dfrac{1}{3^x+3}\,dx$$

Question 2

Question Stem for Question Nos. 15 and 16

Consider the curve $$C_1$$ given by $$y=e^{-x}$$ for $$x\in[0,10\pi]$$, and the curve $$C_2$$ given by $$y=e^{-x}(\sin x+\cos x)$$ for $$x\in[0,10\pi]$$.

Let $$n$$ be the total number of points of intersection of the curves $$C_1$$ and $$C_2$$.

Suppose that $$\alpha_1,\alpha_2,\dots,\alpha_n\in[0,10\pi]$$ are the $$x$$-coordinates of the points of intersection of the curves $$C_1$$ and $$C_2$$ such that $$\alpha_1<\alpha_2<\cdots<\alpha_n$$.

Let $$\beta$$ be the area of the region enclosed between the curves $$C_1$$, $$C_2$$, and the lines $$x=\alpha_1$$ and $$x=\alpha_4$$. Then the value of

$$-\dfrac{1}{\pi}\log_e\!\left(\beta-2\,e^{-\frac{\pi}{2}}\right)$$

is ___.

Question 3

If $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{x^2\cos x}{1+\pi^x}+\frac{1+\sin^2x}{1+e^{\sin x^{2023}}}\right)dx=\frac{\pi}{4}(\pi+a)-2,$$ then the value of $$a$$ is: