$\text { Use integration tables to find } \int \frac { 28 t ^ { 6 } } { 1 + \sin \left( t ^ { 7 } \right) } d t \text {. }$
A) $4 \tan \left( t ^ { 7 } \right) + 4 \sec \left( t ^ { 7 } \right) + C$
B) $28 \tan \left( t ^ { 7 } \right) + 28 \sec \left( t ^ { 7 } \right) + C$
C) $4 \tan \left( t ^ { 6 } \right) - 4 \sec \left( t ^ { 6 } \right) + C$
D) $4 \tan \left( t ^ { 7 } \right) \pm 4 \sec \left( t ^ { 7 } \right) + C$
E) $4 \tan \left( t ^ { 7 } \right) - 4 \sec \left( t ^ { 7 } \right) + C$

Question

$\text { Use integration tables to find } \int \frac { 28 t ^ { 6 } } { 1 + \sin \left( t ^ { 7 } \right) } d t \text {. }$ 
A) $4 \tan \left( t ^ { 7 } \right) + 4 \sec \left( t ^ { 7 } \right) + C$
B) $28 \tan \left( t ^ { 7 } \right) + 28 \sec \left( t ^ { 7 } \right) + C$
C) $4 \tan \left( t ^ { 6 } \right) - 4 \sec \left( t ^ { 6 } \right) + C$
D) $4 \tan \left( t ^ { 7 } \right) \pm 4 \sec \left( t ^ { 7 } \right) + C$
E) $4 \tan \left( t ^ { 7 } \right) - 4 \sec \left( t ^ { 7 } \right) + C$

Quizplus · Accepted Answer

We can use the substitution $u=t^7$ to obtain:
$$\int \frac{28t^6}{1+\sin(t^7)}dt = \frac{4}{7}\int \frac{28u^{\frac{6}{7}}}{1+\sin(u)}du$$
Then, using a table of integrals, we get:
$$\frac{4}{7}\int \frac{28u^{\frac{6}{7}}}{1+\sin(u)}du = \frac{16}{7} \tan\left(\frac{u}{2}+\frac{\pi}{4}\right)+C$$
Substituting back $u=t^7$, we get:
$$\int \frac{28t^6}{1+\sin(t^7)}dt = \frac{16}{7}\tan\left(\frac{t^7}{2}+\frac{\pi}{4}\right)+C$$
Simplifying, we obtain:
$$\frac{16}{7}\tan\left(\frac{t^7}{2}+\frac{\pi}{4}\right)+C = \frac{16}{7}\left(\tan\left(\frac{t^7}{2}\right)+1\right)+C$$
Therefore, the correct answer is $\boxed{\text{(E) }4\tan(t^7)-4\sec(t^7)+C}$.

Use integration tables to find ∫28t61+sin⁡(t7)dt. \text { Use integration tables to find } \int \frac { 28 t ^ { 6 } } { 1 + \sin \left( t ^ { 7 } \right) } d t \text {. } Use integration tables to find ∫1+sin(t7)28t6​dt.

$\text { Use integration tables to find } \int \frac { 28 t ^ { 6 } } { 1 + \sin \left( t ^ { 7 } \right) } d t \text {. }$