integral from-pi/4 to 0 of (tan(x))^2

解

\int_{- \frac{π}{4}}^{0} (tan (x))^{2} d x

1 - \frac{π}{4}

十進法表記

0.21460 \dots

解答ステップ

\int_{- \frac{π}{4}}^{0} (tan (x))^{2} d x

三角関数の公式を使用して書き換える

= \int_{- \frac{π}{4}}^{0} \frac{1 - cos ^{2} ( x )}{cos ^{2} ( x )} d x

拡張

\frac{1 - cos ^{2} ( x )}{cos ^{2} ( x )} : \frac{1}{cos ^{2} ( x )} - 1

= \int_{- \frac{π}{4}}^{0} \frac{1}{cos ^{2} ( x )} - 1 d x

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{- \frac{π}{4}}^{0} \frac{1}{cos ^{2} ( x )} d x - \int_{- \frac{π}{4}}^{0} 1 d x

\int_{- \frac{π}{4}}^{0} \frac{1}{cos ^{2} ( x )} d x = 1

\int_{- \frac{π}{4}}^{0} 1 d x = \frac{π}{4}

= 1 - \frac{π}{4}