Question

\(5 - 32\)Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

7. \(\int_{ - \infty }^0 {\frac{1}{{3 - 4x}}} dx\)

Short answer

Integral is divergent.

Step-by-step solution

Condition for diverging and converging integral.

Consider the integral\(\int_{\bf{a}}^{\bf{t}} {\bf{f}} {\bf{(x)dx}}\)is for the limit with the finite number\(t \ge a\). Then, the integral is written as:

\(\int_a^\infty f (x)dx = \mathop {\lim }\limits_{t \to \infty } \int_a^t f (x)dx\)

Evaluate integral.

Consider that \(\int {\frac{1}{{a + bx}}} dx = \frac{1}{b}\ln |a + bx|\)using this evaluate the given integral.

Plug \(a = 3,b = - 4\) in this formula,

\(\int {\frac{1}{{3 - 4x}}} dx = \frac{1}{{ - 4}}\ln \left| {3 - 4x} \right|\)

Therefore, the value of \(\int_{ - \infty }^0 {\frac{1}{{3 - 4x}}} dx\)is evaluated as:

\(\begin{aligned}{c}\int_{ - \infty }^0 {\frac{1}{{3 - 4x}}} dx = \left\{ {\frac{1}{{ - 4}}\ln \left| {3 - 4x} \right|} \right\}_{ - \infty }^0\left( {{\mathop{\rm since}\nolimits} \int {\frac{1}{{a + bx}}} dx = \frac{1}{b}\ln \left| {a + bx} \right|} \right)\\ = \left\{ {\frac{1}{{ - 4}}\ln \left| {3 - 4\left( 0 \right)} \right|} \right\} - \left\{ {\frac{1}{{ - 4}}\ln \left| {3 - 4\left( { - \infty } \right)} \right|} \right\}\\ = - \frac{1}{4}\ln 3 + \infty \\ = \infty \end{aligned}\)

Since the value of integral is not finite so the integral diverges.