Expert answers to "Evaluate the integrals. The integrals are listed in..."

iohanetc

Answered question

2021-10-22

Evaluate the integrals. The integrals are listed in random order so you need to decide which integration technique to use.

\int \tan 4 x \sec 4 x d x

Leonard Stokes

Skilled2021-10-23Added 98 answers

I = \int \tan 4 x \sec 4 x d x

I = \int \frac{\sin 4 x}{\cos 4 x} \times \frac{1}{\cos 4 x} d x

I = \int \frac{\sin 4 x}{\cos^{2} 4 x} d x

Let

\cos 4 x = t

- 4 \sin 4 x d x = d t

\sin 4 x d x = - \frac{d t}{4}

I = \int \frac{- d t}{4 t^{2}}

I = \frac{- 1}{4} \frac{t^{- 2 + 1}}{- 2 + 1} + C

I = \frac{1}{4 t} + C

I = \frac{1}{4 \cos 4 x} + C

I = \frac{\sec 4 x}{4} + C

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