First slide

Introduction to integration

Question

$\int \frac{8 x + 13}{\sqrt{4 x + 7}} d x = = A (4 x + 7)^{3 / 2} + B (4 x + 7)^{1 / 2} + C t h e n A + B =$

Easy

A

$\frac{1}{6}$
B

$\frac{- 1}{6}$
C

$\frac{1}{3}$
D

$\frac{1}{2}$

Solution

$\begin{array}{l} \int \frac{8 x + 13}{\sqrt{4 x + 7}} d x \\ = \int \frac{8 x + 14 - 1}{\sqrt{4 x + 7}} d x \\ = \int \frac{2 (4 x + 7) - 1}{\sqrt{4 x + 7}} d x \\ = 2 \int \sqrt{4 x + 7} d x - \int \frac{1}{\sqrt{4 x + 7}} d x \\ = 2 \{\frac{(4 x + 7)^{3 / 2}}{4 \times \frac{3}{2}}\} - \{\frac{(4 x + 7)^{1 / 2}}{4 \times \frac{1}{2}}\} + C \\ = \frac{1}{3} (4 x + 7)^{3 / 2} - \frac{1}{2} (4 x + 7)^{1 / 2} + C \end{array}$

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