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Question Number 131527 by Sudip last updated on 05/Feb/21

Answered by Dwaipayan Shikari last updated on 05/Feb/21

$$\frac{2}{7}\int \frac{t}{t+\frac{22}{7}}dt+\frac{5}{7}\int \frac{1}{t+\frac{22}{7}}dt$$$$=\frac{2}{7}t-\frac{44}{49}log(t+\frac{22}{7})+\frac{5}{7}log(t+\frac{22}{7})$$

Answered by mr W last updated on 05/Feb/21

$$\int \frac{2t+5}{7t+22}dt$$$$=\frac{1}{7}\int \frac{2\times 7t+2\times 22-9}{7t+22}dt$$$$=\frac{1}{7}\int (2-\frac{9}{7t+22})dt$$$$=\frac{1}{7}(2t-\int \frac{9}{7t+22}dt)$$$$=\frac{1}{7}[2t-\frac{9}{7}\int \frac{1}{7t+22}d(7t+22)]$$$$=\frac{1}{7}[2t-\frac{9}{7}\ln (7t+22)]+C$$$$=\frac{2}{7}t-\frac{9}{49}\ln (7t+22)+C$$

Answered by Bird last updated on 06/Feb/21

$$I=\int \frac{2t+5}{7t+22}dtwedothech.7t+22=x\Rightarrow$$$$t=\frac{x-22}{7}\Rightarrow I=\int \frac{1}{x}(\frac{2x-44}{7}+5)\frac{dx}{7}$$$$=\frac{1}{49}\int \frac{1}{x}(2x-44+35)dx$$$$=\frac{1}{49}\int \frac{2x-9}{x}dx$$$$=\frac{2}{49}x-\frac{9}{49}ln\mid x\mid +C$$$$=\frac{2}{49}(7t+22)-\frac{9}{49}ln\mid 7t+22\mid +C$$

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