91Ó°ÊÓ

Find the area under the curve \(y=1 / \cos ^{2} t\) between \(t=0\) and \(t=\pi / 2\).

Find the integrals. Check your answers by differentiation. $$\int \frac{1+e^{x}}{\sqrt{x+e^{x}}} d x$$

Complete the square and give a substitution (not necessarily trigonometric) which could be used to compute the integrals. $$\int(t+2) \sin \left(t^{2}+4 t+7\right) d t$$

Give an example of: A positive, continuous function \(f(x)\) such that \(\int_{1}^{\infty} f(x) d x\) diverges and $$ f(x) \leq \frac{3}{7 x-2 \sin x}, \quad \text { for } x \geq 1 $$

Find the integrals. Check your answers by differentiation. $$\int \frac{e^{x}}{2+e^{x}} d x$$

explain what is wrong with the statement. The midpoint rule never gives the exact value of a definite integral.

Decide whether the statements are true or false. Give an explanation for your answer. The integral \(\int_{0}^{\infty} \frac{1}{e^{x}+x} d x\) converges.

Evaluate the integrals, both exactly [e.g. \(\ln (3 \pi)] \text { and numerically [e.g. } \ln (3 \pi) \approx 2.243].\) $$\int_{0}^{5} \ln (1+t) d t$$

Complete the square and give a substitution (not necessarily trigonometric) which could be used to compute the integrals. $$\int(2-\theta) \cos \left(\theta^{2}-4 \theta\right) d \theta$$

Anti differentiate using the table of integrals. You may need to transform the integrals first. $$\int \sin ^{3} 3 \theta \cos ^{2} 3 \theta d \theta$$