What is the integral of sec^2(t)tan^7(t) ?

The integral of sec^2(t)tan^7(t) is (tan^8(t))/8+C

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integral of sec^2(t)tan^7(t)

\int sec^{2} (t) tan^{7} (t) d t

\frac{tan ^{8} ( t )}{8} + C

Solution steps

\int sec^{2} (t) tan^{7} (t) d t

Rewrite using trig identities

= \int (1 + tan^{2} (t)) tan^{7} (t) d t

Apply u-substitution

: \int u^{7} d u

= \int u^{7} d u

Apply the Power Rule

: \frac{u ^{8}}{8}

= \frac{u ^{8}}{8}

Substitute back

u = tan (t)

= \frac{tan ^{8} ( t )}{8}

Add a constant to the solution

= \frac{tan ^{8} ( t )}{8} + C

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What is the integral of sec^2(t)tan^7(t) ?
The integral of sec^2(t)tan^7(t) is (tan^8(t))/8+C