Giải bài tập 6 trang 42 SGK Toán 12 tập 2 - Cánh diều — Không quảng cáo

Đề bài

Tính

a) $\int\limits_{ - 1}^1 {{{(x + 2)}^3}} dx$

b) $\int\limits_1^2 {\frac{2}{{{x^2}}}} dx$

c) $\int\limits_1^4 {{x^2}\sqrt x } dx$

d) $\int\limits_{ - 1}^0 {{2^{3x + 2}}} dx$

e) $\int\limits_0^2 {{2^x}{{.3}^{x + 1}}} dx$

g) $\int\limits_0^1 {\frac{{{7^x}}}{{{{11}^x}}}} dx$

Lời giải chi tiết

a) $\int\limits_{ - 1}^1 {{{(x + 2)}^3}} dx = \left. {\frac{{{{(x + 2)}^4}}}{4}} \right|_{ - 1}^1 = 20$

b) $\int\limits_1^2 {\frac{2}{{{x^2}}}} dx = \left. { - \frac{2}{x}} \right|_1^2 = 1$

c) $\int\limits_1^4 {{x^2}\sqrt x } dx = \int\limits_1^4 {{x^{\frac{5}{2}}}} dx = \left. {\frac{2}{7}{x^{\frac{7}{2}}}} \right|_1^4 = \frac{{254}}{7}$

d) $\int\limits_{ - 1}^0 {{2^{3x + 2}}} dx = \left. {\frac{{{2^{3x + 2}}}}{{3.\ln 2}}} \right|_{ - 1}^0 = \frac{4}{{3\ln 2}} - \frac{1}{{6\ln 2}}$

e) $\int\limits_0^2 {{2^x}{{.3}^{x + 1}}} dx = \int\limits_0^2 {{6^x}.3} dx = \left. {\frac{{{{3.6}^x}}}{{\ln 6}}} \right|_0^2 = \frac{{105}}{{\ln 6}}$

g) $\int\limits_0^1 {\frac{{{7^x}}}{{{{11}^x}}}} dx = \left. {\frac{{{{\left( {\frac{7}{{11}}} \right)}^x}}}{{\ln \frac{7}{{11}}}}} \right|_0^1 = \frac{{ - 4}}{{11\ln \frac{7}{{11}}}}$