Question
Evaluate the following definite integrals:
$\int_{-2}^\limits{3}\frac{1}{\text{x}+7} \text{ dx}$
$\int_{-2}^\limits{3}\frac{1}{\text{x}+7} \text{ dx}$
✓
Answer
We know that $\int\frac{\text{dx}}{\text{x}}=\log\text{x}+\text{C}$
Now,
$\int_{-2}^\limits{3}\frac{1}{\text{x}+7} \text{ dx}$
$=\big[\log(\text{x}+7)\big]^3_{-2}$
$=\big[\log10-\log5]^3_{-2}$
$=\log\frac{10}{5}$ $\Big[\because\log\text{a}-\log\text{b}=\log\frac{\text{a}}{\text{b}}\Big]$
$=\log2$
$\therefore\ \int_{-2}^\limits{3}\frac{1}{\text{x}+7} \text{ dx}=\log2$
Now,
$\int_{-2}^\limits{3}\frac{1}{\text{x}+7} \text{ dx}$
$=\big[\log(\text{x}+7)\big]^3_{-2}$
$=\big[\log10-\log5]^3_{-2}$
$=\log\frac{10}{5}$ $\Big[\because\log\text{a}-\log\text{b}=\log\frac{\text{a}}{\text{b}}\Big]$
$=\log2$
$\therefore\ \int_{-2}^\limits{3}\frac{1}{\text{x}+7} \text{ dx}=\log2$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Which one of the following graphs represents a function?
If $\vec{\text{a}}$ and $\vec{\text{b}}$ are two vectors of the same magnitude inclined at an angle of $60^\circ$ such that $\vec{\text{a}}.\vec{\text{b}}=8,$ write the value of their magnitude.
→Find the equation of the plane passing through the following point:
(1, 1, 1), (1, -1, 2) and (-2, -2, 2)
(1, 1, 1), (1, -1, 2) and (-2, -2, 2)
Find: $\int\sin^{-1}(2\text{x})\text{dx}.$
→A company produces two types of goods A and B, that require gold and silver. Each unit of type A requires 3 g of silver and 1 g of gold while that of type B requires 1 g of silver and 2 g of gold. The company can procure a maximum of 9 g of silver and 8 g of gold. If each unit of type A brings a profit of ₹ 40 and that of type B ₹ 50, formulate LPP to maximize profit.
Evaluate the determinant $\Delta=\left|\begin{array}{rrr} {1} & {2} & {4} \\ {-1} & {3} & {0} \\ {4} & {1} & {0} \end{array}\right|$
→Determine whether the following operations define a binary operation on the given set or not:$'\odot'$ on N defined by $\text{a}\odot\text{b}=\text{a}^{\text{b}}+\text{b}^{\text{a}}$ for all $\text{a, b}\in\text{N.}$
→Given two independent events A and B such that P(A) = 0.3 and P(B) = 0.6. Find
$\text{P}(\overline{\text{A}}\cap\overline{\text{B}})$
→$\text{P}(\overline{\text{A}}\cap\overline{\text{B}})$
If $\vec{\text{a}},\vec{\text{b}},\vec{\text{c}}$ are the position vectors of the vertices of an equilateral triangle whose orthocentre is at the origin, then write the values of $\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}$.
→Find the value of c in Rolle’s theorem for the function $\text{f(x)} = \text{x}^{3} - \text{3x in} [ -\sqrt{3}, 0].$
→