If x 6 sin ( 5 x 7 ) ? d x = k 5 cos ( 5 x 7 ) + C , x 0 , then:

Question

If $\int x^{6} \sin (5 x^{7}) d x = \frac{k}{5} \cos (5 x^{7}) + C, x \neq 0,$ then:

Smart Achievers · Accepted Answer

(4)

Ans. $k = - \frac{1}{7}$

Explanation:

Put $5 x^{7} = t$

$\Rightarrow$ $35 x^{6} d x = d t$

$\Rightarrow$ $x^{6} d x = \frac{d t}{35}$

Therefore, $\int x^{6} \sin (5 x^{7}) d x = \int \sin t \cdot \frac{d t}{35}$

$= - \frac{\cos t}{35} + C$

$= - \frac{\cos (5 x^{7})}{35} + C$

Comparing with

$\int x^{6} \sin (5 x^{7}) d x = \frac{k}{5} \cos (5 x^{7}) + C,$

we get $\frac{k}{5} = - \frac{1}{35}$

$∴$ $k = - \frac{1}{7}$

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