【高校数学Ⅲ】5-2-3 定積分と導関数｜要点まとめ

\(f'(x)=x^3e^{x^2}\)

\(\displaystyle =\int_0^x (xe^t-te^t)dt\)
\(f'(x)=(\int_0^x xe^tdt)'-xe^x\)
\(\displaystyle =\int_0^x e^tdt+x(\int_0^x e^tdt)'-xe^x\)
\(=e^x-e^0+xe^x-xe^x\)
\(=e^x-1\)

\(\displaystyle a=\int_0^{\frac{\pi}{3}} f(t)\sin tdt\)とおくと、
\(f(x)=\cos x+a\)
\(f(t)=\cos t+a\)
\(\displaystyle a=\int_0^{\frac{\pi}{3}} (\cos t+a)\sin tdt\)
\(\displaystyle a=\int_0^{\frac{\pi}{3}} \cos t\sin tdt+a\int_0^{\frac{\pi}{3}} \sin tdt\)
\(\displaystyle a=\int_0^{\frac{\pi}{3}} \frac{1}{2}\sin2tdt+a[-\cos t]_0^{\frac{\pi}{3}}\)
\(\displaystyle a=-\frac{1}{4}[\cos2t]_0^{\frac{\pi}{3}}+\frac{1}{2}a\)
\(\displaystyle \frac{1}{2}a=\frac{3}{8}\)
\(\displaystyle a=\frac{3}{4}\)
よって、
\(\displaystyle f(x)=\cos x+\frac{3}{4}\)